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The Teaching of Mathematics 


tn the 


bigber Schools of Prussia 





The 
Ceaching of Mathematics 


in tbe 


higher Schools of Prussia 


BY 


J. W. A. YOUNG, PuD.. 


ASSISTANT PROFESSOR OF THE PEDAGOGY OF MATHEMATICS 
IN THE UNIVERSITY OF CHICAGO 


LONGMANS, GREEN, AND CO. 
gt AND 93 FIFTH AVENUE, NEW YORK 
LONDON AND BOMBAY 
1900 





ATE NSE UB Bd Se a ee 
Vols Ty to Ve baieke bal 
; 7 Ae ae et 






CopyRIGHT, goo, BY 
LONGMANS, GREEN, AND CO, — 





All rights reserved 


THE CAXTON PRESS 
oe NEW YORK. 
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; ‘ : 2 : wae x a 2H 
, ce ice. 
+ ot Siders ee LIE 








2300t-Y 


, 


Preface 


The well-known fact that the Prussians have 
long been studying the problems of education sys- 
tematically, thoroughly, and successfully, led me 
vecently to spend nearly an entire academic year 
in examining the outcome of their study as evinced 
in the present status of the work of education in 
Prussia. Some of the results of this examination 


°5 are presented herewith. 


It was my primary aim in examining the Prus- 


_ stan higher-school system to ascertain their meth- 


ods of teaching mathematics, but the work in 
mathematics cannot be understood without some 
acquaintance with the entire system of which the 
mathematical work ts a part and by whose spirit zt 
ts dominated. Consequently the following account 
combines a general sketch of the Prussian higher- 
school system with a more specific and detailed 
description of the work in mathematics. 

The reader who desires to learn more of the 


vi | Preface 


German higher schools in general than is contained 
in the condensed outline which suffices for the pur- 
poses of the present report, ts referred to the work 
of Russell} on the German higher schools. 

The material for the following account has been 
collected in part from the official and other publi- 
cations relative to these schools, and in part from 
many visits of observation to the institutions them- 
selves. The permit requisite for these visits was 
secured for me from the Minister of Education 
through the intermediation of His Excellency, the 
American Ambassador at Berlin, to whom I feel 
deeply grateful for this kindness. I met with the 
utmost courtesy in all my visits, the various 
Directors and instructors were most obliging and 
did all in their power to assist me in gaining the 
information of which I was in quest, and I wish 
to express my most hearty appreciation of their 
hospitable attentions. 

My thanks are also due to Professor H. Maschke, 


1 Russell, German Higher Schools: The History, Organization 
and Methods of Secondary Education in Germany. Longmans, 
Green & Co., 1898, pp. 455. This work has come to my hands 
since the following pages were prepared, and I have read it with 
great interest. It is thorough and scholarly as well as interesting, 
and I know of no other work in English so complete, satisfac- 
tory, and reliable. 


Preface vii 


of the University of Chicago, formerly Professor 
in the Luisenstaddtische Gymnasium, Berlin, Mr. 
C. E. Linebarger, Instructor in the Lake View 
Fligh School, Chicago, and Mr. E. R. Breslich, a 
graduate of a Prussian gymnasium, for reading 
the following pages in manuscript, critically from 
their respective stand-points. 


F. W. A. Youne. 





Contents 


PAGE 
PREFACE, . . : : : : : 4 3 ny 
I. INTRODUCTION, : : ‘ ‘ : ae | 
Why study Prussian results, 1 
A grave disparity, I 
Possibility of comparison, I 
The ratio of seven to four stated, 2 
Questions raised, 3 
What is to be learned, 3 
II. THE GENERAL STATUS OF THE HIGHER SCHOOLS, 5 


Common vs. higher schools, 5 
Education a duty or a privilege, 6 
The types of higher schools, 8 


III, THE GOVERNMENT, 


The King, 9 

The Minister of Education, 9 
The provincial school-board, 9 
Circular orders, 9 

The Director, 11 

The ordinary, 11 


x Contents 


PAGE 
IV. THE TEACHERS, ; : e . ° . Ung 
Preparation, 13 
Examination, 13 
By whom conducted, 13 
Requirements for admission, 14 
The subjects, 14 
The scope, 15 
General, 15 
Special attainments in mathematics, 15 
For middle classes, 15 
For upper classes, 16 
The mode of the examination, 16 
Written, 16 
rai 3 
Results of the examination, 17 
The seminary year, 18 
Purpose and nature, 18 
The work, 18 
Weekly conferences, 19 
Observation of instruction, 20 
Giving instruction, 20 
The trial year, 21 
Summary of preparation, 22 
Appointment, 23 
Ranks and honors, 23 
Duties, 24 
Salaries, 25 
Pensions, 26 
The ‘* jubilee,” 27 
Total income, including estimated equivalent of 
pensions, 28 
Purchasing power in America, 30 
The effect of the pension system; tranquillity of 
life, 31 


Contents xl 


PAGE 
eehete EOPILS,~ .«. : : : ° ° : SRK: 
Admission, 33 
Classification and age, 33 


VI. THE INSTITUTIONS, . a ee tear . ° sue a0 


Classification, 36 
Number, 37 
Financial support, 38 
Buildings, 38 
Equipment, 39 
Faculty-room, 39 
Sessions, 40 

The hourly pause, 41 


VIL. THE CURRICULA, ; ‘ ° 4 : nA 


First curriculum, 43 
Conference of 1890, 43 
Detailed curriculum of gymnasium, 45 
Synopsis of curricula of present century, 46 
The curriculum in mathematics for the gymna- 
sium, 47 
General aim of the instruction, 47 
Scope of the instruction, 47 
Methodic remarks, 50 
The curriculum in mathematics for the Realgymna- 
sium and for the Oberrealschule, 51 
German criticism of the methodic directions, 52 
Distribution of the hours, 52 


xil Contents 


PAGE 
VIII. THE INSTRUCTION IN MATHEMATICS, . ; «tae 


Classroom customs, 53 
The size of the classes, 53 
Teaching, not ‘‘ hearing recitations,” 54 
Stress on the class-exercise, 54 
‘‘Socratic method,” 55 _ 
Oral work, 57 
Diarium, 58 
‘« Chalk and talk,” 59 
A lesson in algebra, 60 
A lesson in geometry, 64 
Objections considered, 67 
The manner of the teachers, 68 
Written exercises; neatness, 69 
Number of teachers in mathematics during the 
course of a pupil, 70 
Homogeneity of instruction, 70 
The class-book, 71 
Specimen entries, 72 
Private study, 73 
Specimen of allotment, 74 
Text-books in mathematics, 75 
Purpose of the text, 77 


IX. THE EXAMINATIONS, ° ; ° ° ° e 79 


Annual, 79 
Final, 80 
Written, 81 
Setting the papers, 81 
Independent work required, 82 
Correcting the papers, $2 
Determining the grade, 83 
Oral, 83 
Privileges of the graduate, 85 


Contents Xill 


X. THE PROGRAMM, . ° . . ° A . 86 
The scientific paper, 86 
The school-report, 86 
The curriculum, 86 
The hours of instruction, 86 
The class-work, 87 
Examination papers in mathematics, 89 
List of text-books, g2 
Orders of the superior boards, 93 
Specimen orders, 93 
Chronicles of the institution, 95 
Statistics, 96 
Additions to libraries, laboratories, and muse- 
ums, 96 
Stipends and funds, 96 
Notices to parents and pupils, 97 


XI. THE REFORMSCHULE, . i i : ; aoe 
Character and purpose, 98 
Detailed curriculum of the Reformschule at Han- 
over, IOI 
Synopsis of various curricula, 102 


XII. THE HIGHER EDUCATION OF WOMEN, . : ialOs 


No gymnasia for women, 103 
What is done, 103 

The quality of the work, 104 
Progress being made, 105 


XIII. CoMPARISON BETWEEN GERMAN AND AMERICAN 
WoRK, F ‘ 2 ‘ : : . 106 
Basis of comparison, 106 
Sketch of American course, 107 
Work done compared, 108 
Time ratios compared, 109 
Comparative table of fractions of time given to math- 
ematics, 110 


X1V Contents 


PAGE 
XIV. CONCLUSION, . : : : , . ‘ . 112 
The disparity, how not caused, 113 
Causes contributory to Prussian excellence, 114 
Central legislation and supervision, 115 
Expert government, I15 
Other conditions, 116 
A desideratum, 116 
Uniformity of curricula, 117 
Preparation of the curricula, 118 
Supervision, 118 
American conditions, 119 
A serious loss, 119 
Resolution IV. of the Committee on College 
Entrance Requirements, 120 
What can be done now, 121 
The preparation and status of the teachers, 122 
Need of pedagogic training, 122 
Raising of standards, 124 
Enthusiasm and devotion needed, 126 
The methods of instruction, 127 
The stress laid on classroom work, 127 
Work under instruction, 128 
Resolution XIV. of the Committee on Col- 
lege Entrance Requirements, 129 
More instruction needed, 130 
The pause, 132 
The distribution of the mathematical subjects 
in the course, 132 
Three points of difference, 132 
Scope of arithmetic, 133 
Order of beginning, 134 
Rate of continuing, 134 
Gradual growth needed, 134 
The method of instruction in mathematics, 136 
The genetic method, 136 
Class vs. individual, 137 
The heuristic method, 138 
Professor Miinsterberg’s views, 140 


THE TEACHING OF 
MATHEMATICS IN PRUSSIA 


I 


TMntroduction 


The teaching of mathematics in the higher 
schools of Prussia deserves the serious atten- 
tion of those interested in the teach- Prieaad 
ing of mathematics in America, not oe 
only because it is the fruit of long labors by a 
nation that has stood and still stands in the 
forefront of educational progress, but also be- 
cause a comparison between the work accom. 
plished in mathematics in Prussia and in the 
United States reveals a disparity of a character 
so grave that American educators cannot afford 
to pass it by unheeded. 

There are very few subjects in which a com- 
parison of the quantity and quality of work ac- 
complished under different curricula pogsipitity of 
and methods can be instituted in any comparison. 
but the most general way. The result is af- 
fected by so many elements which cannot be 

I 


2 Teaching of Mathematics in Prussia 


specified in black and white, or tested by exe 
aminations, that it is wellnigh impossible to 
find a satisfactory standard of comparison. In 
mathematics, however, this is the case to a 
much smaller extent than in other branches. 
The subject-matter of school mathematics has 
long since been so systematized, and its nature 
permits so little variation in the topics taken up 
and the order of their consideration, that the 
quantity of work done may be quite clearly de- 
scribed by a list of topics and its quality suf- 
ficiently well tested by examinations. 

The present study of the Prussian secondary 
school system was begun under the impression 
Ratio of (quite current in America) that, while 
seven to four. the work of the Germans is perhaps 
more thorough, it is accomplished with a great- 
er outlay of time than is devoted to the same 
subject-matter in this country. But compari- 
son of curricula and time-schedules reveals the 
startling fact (which will be substantiated in de- 
tail in the sequel) that zz the work in mathematics 
done in the nine years from the age of nine on, we 
Americans accomplish no more than the Prussians, 
while we give to this work about seven-fourth (1.72) 
tomes as large a fraction of the total time of in- 
struction as do the Prussians. 

In other words, the Prussians give about 1.2 
years of the nine years in question to mathematics, 
accomplishing fully as much as the Americans, who 


Tntroduction 3 


give about 2.1 years to the same work—a differ- 
ence of nine-tenths of a year, or one-tenth of the 
total time of instruction in these nine years. 

This state of affairs is certainly one which 
demands most careful consideration at our 
hands. Is there really so great adis- questions for 
parity as appears on the face of the ‘consideration, 
time-schedules? Is there in reality any disparity 
at all? Ifso, to whatisit due? Is it possible 
for Americans so to modify their system and 
methods as to diminish the disparity? What 
lessons can Americans learn from the Prussian 
system? These and other questions suggest 
themselves at once, and it is with these ques- 
tions in the foreground that the following ac- 
count of the Prussian system should be read. 

The writer wishes to say at the outset that, 
while he believes that there is a real disparity 
and that it ought to be lessened, he whatisto 
by no means advocates that the Prus- __ be learned. 
sian system as such be adopted or imitated here. 
It does seem, however, that the indisputable su- 
periority which the facts mentioned above show 
the Prussian system to have in its own environ- 
ment, over the American system in its environ- 
ment must oblige American educators to study 
the Prussian system most carefully, especially 
along its lines of divergence from their own. 
While the outcome of such study may be that 
little or nothing is found which we can directly 


4 Teaching of Mathematics in Prussia 


adopt, hints may perhaps be gleaned which we 
may adap? to our own circumstances with signal 
profit. Education is more a problem of human- 
ity than of nationality, and while distinctively 
German methods might not prove strong else- 
where, those results which the Germans have 
attained as men and not as Germans must be of 
great significance the world over. 


I] 
The General Status of the higber Schools 


The distinction between the common and the 
higher schools (Volks- und hihere Schulen) must 
be noticed at the very outset. Thisis (| 
not a distinction in any way analo- vs. nigher 
gous to our grades and high schools, *"°'* 
but constitutes a complete differentiation of 
the boys from almost or quite the beginning 
of their school career into two distinct classes. 
In the Volksschulen the aim is to train good and 
faithful citizens ; the process is called Erziehung 
(“bringing up”). In the higher schools, on 
the other hand, the aim is to impart learning 
and to turn out men who are educated or cult- 
ured (gedz/det); the process is called Unterricht 
(instruction) and leads to privileges and respon- 
sibilities before the civil and the military law, 
and the unwritten social law as well. The 
higher schools proper take the boys at the age 
of nine and have a curriculum covering nine 
years ; in many cases a preparatory school with 
a course of three years is connected with the 
institution, so that a boy of six years may step 

5 


6 Teaching of Mathematics in Prussia 


into the work of a higher school. As the com- 
mon schools do very little in mathematics, only 
the work of the higher schools will be con- 
sidered in what follows. Several American 
writers have already given general descrip- 
tions of the German common school system, to 
which the reader who may be interested in the 
work of these schools is referred.’ 

A thorough German compend covering 
the entire educational system is that of Peter- 
silie,? in which all grades of institutions from 
the Universities down are described and their 
regulations collated. This is done in consider- 
able detail for Germany, and in a summary 
manner for the other principal countries of 
Europe. 

In Prussia the state regards attendance upon 
a higher school as a previlege ; for the common 
Education 00d the state may restrict the num- 
adutyora ber of persons admitted to such at- 
privilege. tendance. But attendance upon the 
common schools is regarded as the duty of 
those not having better opportunities, and is 
enforced by the state. The attendance upon 


1 For example, Klemm, Zuvopean Schools, D. Appleton & Co., 
1889; Seeley, Zhe Common Schools of Germany, Kellogg & Co., 
1896. 

® Petersilie, Das offentliche Unterrichtswesen im deutschen Reiche 
und in den tibrigen europdischen Kulturlindern, Leipzig, 1897, 
2 Bde., pp. 448, 608. 





The General Status of the thigher Schools 7 


higher schools is now being restricted by the 
state by the simple expedient of founding fewer 
new institutions than would be adequate to 
meet the present demand for admittance, which 
is far in excess of the number that can be re- 
ceived by the institutions now in existence. 
This is done to abate the crying evil which 
Bismarck called the Adzturzentenproletariat 
(“beggar-graduates”). The graduate of a 
higher school is admitted to occupations closed 
to all others, and the social usages are such 
as to prevent him, on pain of losing caste, from 
entering any of another large group of occupa- 
tions. Consequently the occupations which are 
considered suitable for graduates are terribly 
overcrowded, and since the pressure cannot be 
relieved by overflow into other occupations, it 
must be relieved at the source of supply. The 
condition of affairs is graphically illustrated by 
an experience of an American resident of Ber- 
lin. An educated German called in silk hat 
and gloves to deg. “Why don’t you work?” 
asked the common-sense American, and was 
met with the indignant reply, “ That would not 
be in keeping with my social station” (“ Das 
ware nicht standesgemiss’’). 

The higher schools are divided into three 
types: the Gymnasium, with both Latin and 
Greek; the Realgymnasium, with Latin but no 
Greek; and the Oderrealschule, with neither 


8 Teaching of Mathematics in Prussia 


Latin nor Greek. The characteristics of these 
institutions will be described later on, this 
Thetypes  Prief mention sufficing for the present. 
ofhigher The corresponding work in Amer- 
re aes ica is divided between institutions of 
differing character, and there are here no 
single institutions analogous to the German 
higher schools; we shall therefore be obliged 
to retain the German names in speaking of 
these schools. 





III 
The Government 


The primary source of educational authority 
in Prussia is the king. The present king (Em- 
peror William II.) takes an active in- he govern- 
terest in the work and was himself _ ing bodies. 
educated in a gymnasium. The head of the 
actual educational work is the Minister of Spir- 
itual, Educational, and Medicinal Affairs (Gezst- 
licher, Unterrichts- und Medtzinal-Angelegenhet- 
ten), who is appointed by the king and is a 
member of the royal cabinet. He is assisted 
by over twenty active councillors for educa- 
tional matters (vortragende Rathe). The Minis- 
ter, in turn, appoints a school-board (Provin- 
gtal-Schulkollegium) for each of the thirteen 
provinces of Prussia, to whom the detailed su- 
pervision of the schools is intrusted. 

Besides particular communications to the 
separate institutions, both the Ministry and 
the provincial boards issue frequent Circular 
circular orders or general bulletins oreerss 
(Circularver figungen) to the institutions respec- 
tively under their charge. These take up on 

9 


10 Teaching of Matbematics in Prussia 


occasion matters of pedagogic method, and of 
administration even in detail, and thus tend to 
produce great uniformity in the work of all the 
institutions of the kingdom. The orders of in- 
terest to the general public are usually pub- 
lished in the annual announcement (Programm) 
of the school, and in treating of the latter, cita- 
tions will be made. 

All the orders since the beginning of the 
century which are still in force have been col- 
lected in the work of Wiese-Kiibler,! which is 
officially recognized, and upon which some of 
the statements of this paper are based. The 
orders are well grouped and indexed, and the 
work as a whole constitutes a complete and 
authoritative exposition of the organization 
and regulations of the Prussian higher school 
system. 

The provincial school-board is composed of 
picked men of experience in school-work, who 
The provine Gevote their entire time to the duties 
cialschool- of this position and who receive the 
see) highest salary paid. Each member 
has a number of institutions assigned to him 
for special personal supervision; he keeps in 
close touch with each, and informed as to its 
work by personal communication with the Di- 

1 Wiese-Kiibler, Verordnungen und Gesetze fiir die hohere 


Schulen in Preussen, 3te Aufl., Bd. I., 1886, pp. 488; Bd. IL., 
1888, pp. 521. 


The Government if 


rector, by visitations (for inspection of instruc. 
tion and conferences with the teachers), and by 
conducting examinations. He is the connect- 
ing link between the school-board as a whole 
and the institutions under his especial charge. 
The Schulkollegium of the province Branden- 
burg, in which Berlin lies, has commodious 
and well-fitted quarters (three floors of a 
large building) and a considerable clerical 
force. 

The head of the institution is the Director, 
appointed by the provincial school-board with 
royal approval. He administers the ne schools 
affairs of the institution, assigns to eee 
each teacher his work, and appoints an “ ordi- 
nary” (Ordinarius) for each class. These are, 
as it were, subdirectors supervising the work 
of single classes as the director does that of the 
entire institution. The ordinary is usually the 
teacher who gives the largest number of hours 
of instruction to the class, and consequently 
the teachers of Latin are sure to be among those 
who are called on to perform this duty. The 
ordinary has charge of all the routine super- 
vision of the class as a whole, and comes into 
closer personal contact with the pupils than do 
the other teachers. 

Everything not strictly in the usual course 
of the work of instruction of a class must be 
referred to the ordinary. Every question not 


12 Teaching of Mathematics in Prussia 


strictly in the usual course of administration of 
a single class must be referred to the director, 
and so on, each officer referring to a higher au- 
thority all questions not falling within the scope 
of his own well-defined powers. 


IV 
The Teachers 


The first step toward becoming a teacher in 
the higher schools of Prussia is the acquisition 
of a liberal education and of sufficient 
scientific attainments in the subject 
which the candidate wishes to teach. It is 
requisite that he have completed the course of 
a gymnasium (for certain subjects that of a 
realgymnasium will suffice), and that he have 
studied three years in German Universities. 
The adequacy of his preparation is tested by 
an examination. 

This is conducted by a board (Kénigliche wts- 
senschaftliche Priifungscommission), appointed for 
that purpose by the Minister of Edu- the exam. 
cation. There are ten such boards in pb 
the kingdom, one board serving for two prov- 
inces in a few cases. Their seat is always in a 
University town, and their membership is made 
up almost entirely of University professors. 

This examination is known as the examina- 
tion pro facultate docendi, or the “ Staatsexa- 
men.” To be admitted to it the candidate 
must submit 


Preparation. 


ee 


14 Teaching of Mathematics in Prussia 


a. A certificate of maturity (equivalent to 
our diploma of graduation) from a German 
gymnasium; if the principal subjects (see be- 
low) are taken from the following: Mathemat- 
ics, natural sciences, foreign modern languages, 
the candidate may offer a certificate of ma- 
turity from a realgymnasium. 

6. Documents to show that he has studied 
three years in a German University (of these 
at least one and one-half years in a Prussian 
University) ; if one of the subjects is English or 
French, the candidate may by special permis- 
sion replace one year’s University study by 
study of the language in question in an institu- 
tion or ina country in which the language is 
spoken. 

The candidate specifies the particular sub- 
jects in which he seeks authorization to teach 
and the grade in which he wishes to 
obtain the teacher’s certificate. There 
are three grades—lower, middle, and upper— 
each constituting three years of the nine years’ 
course. He must offer at least two principal 
and two subordinate subjects. These subjects 
are selected from a list which is practically that 
of the subjects taught in the gymnasia (see cur- 
ricula below), the combinations being subject 
to a few restrictions which are of little conse- 
quence in the present paper. 

The scope of the examination is twofold, 


The subjects. 


The Teachers 15 


testing the candidate’s fitness for the post of 
teacher in general and in particular. The spe- 
cific requirements are: 

1. General.—All candidates are examined in 
philosophy and pedagogy, the German lan- 
guage and literature, and, if Chris- tibet: 
tians, in the contents of Holy Script- _ theexami- 
ure, Church history, and the dogmas aa 
of that church (State Church or Roman Catho- 
lic) to which they belong. The object of this 
part of the examination is to determine whether 
the candidate possesses that general culture 
which is to be demanded of all instructors in 
higher schools. 

The examination is rigorous, and as it may 
take up topics from a wide field, and as the ex- 
aminer in each subject is usually a University 
professor and always a specialist in the subject, 
the candidates often anticipate this examina- 
tion with more apprehension than that in the 
special subjects which they wish to teach. 

2. Special Attainments.—To obtain the teach- 
er’s certificate (Oberlehrerzeugniss), the candi- 
date must obtain the authorization (Befahigung) 
to teach two (principal) subjects in all classes 
and two (subordinate) subjects in the middle 
classes. For the subject of mathematics the 
scope of the examination is as follows: 

a. For Middle Classes.—Plane and solid geom- 
etry, algebra through quadratics, logarithms, 


16 Teaching of Matbematics in Prussia 


properties of the decimal system of numeration, 
equations of the third and the fourth degree, 
spherical trigonometry with applications to 
mathematical geography, plane analytic geom- 
etry, and the elements of the differential and 
integral calculus. 

b. For Upper Classes.—In addition to the fore- 
going the candidate must show that he possesses 
such acquaintance with the most important 
branches of higher geometry, analysis, and an- 
alytic mechanics as will enable him to treat in- 
dependently a not too difficult problem from 
one of these fields, and he must be acquainted 
with the more important literature of these 
subjects. 

The examination consists of two parts: the 
written examination, in which the candidate 
The mode of Prepares papers privately on assigned 
examination. tonics, and the oral examination, in 
which he appears before the Commission in 
person and may be examined at will by each 
member. 

» I. The Written Examination.—Subjects are 
assigned to the candidate, one from each of his 
principal subjects, also one from any subordi- 
nate subject in which he may wish to obtain the 
authorization to teach in all classes, and one 
from philosophy or pedagogy. Not more than 
three subjects may be assigned altogether. The 
candidate prepares at home a paper on each 


The Teachers . 17 


subject assigned. Those from classical phi- 
lology are treated in Latin, those from foreign 
modern languages in the language concerned; 
all others in German, except by special permis- 
sion. 

Eight weeks’ time is allowed for the prep- 
aration of each paper (an additional eight 
weeks may be granted upon due application, 
and still more if necessary), and at the close of 
the total time accruing for all subjects all the 
papers are handed in, with the candidate’s as- 
surance that they were prepared by himself 
with no other assistance from persons or books 
than that specified in detail by him. 

If the candidate submits a printed paper 
written by himself, it may be accepted provided 
the subject and contents are satisfactory to the 
board. If the paper have been approved by 
the faculty of a Prussian University as a dis- 
sertation for the Doctor’s degree, only the sub- 
ject of the paper is scrutinized by the examin- 
ing board. 

2. The Oral Examination.—If the results of 
the written examination have been sufficiently 
good, the candidate is notified to appear for the 
oral examination which extends over all the 
subjects offered by him, and includes the gen- 
eral culture topics as well. 

Results of the Examination.—The candidate 
may be passed unconditionally, conditionally, 


18 Teaching of Mathematics in Prussia © 


or rejected. In appropriate cases there are 
open to him repetition of the examination, 
supplementary examinations (to work off con- 
ditions), and additional examinations (in new 
subjects for extension of teaching privileges). 

After it has been ascertained by the examina- 
tion that the candidate is possessed of liberal 
Thesemi. Culture and of sufficient specific sci- 
nary year. entific attainments in the subjects he 
wishes to teach, he must next devote a year 
(Seminarjahr) to the study of the art of teach- 
ing with a view to the practical exercise of his 
profession. 

To give opportunity for this study, pedagogic 
seminaries have recently been organized in 
connection with various ones of the schools, 
and to these the candidates are assigned in 
numbers not to exceed six for each semi- 
nary. 

The director of the institution conducts the 
work of the seminary, and the seminaries 
have been located in institutions whose direc- 
tors are men eminent in the pedagogic world. 
The work of the seminarist is of three sorts: 

1. Weekly conferences of the seminary. 

2. Observation of teaching. 

3. Teaching under supervision and guid- 
ance. 

The weekly conference is held under the 
presidency of the director, and an experienced 


The Teachers 19 


professor of the subject which is at the time 
under special consideration also attends the ses- 
sion and participatesinits work. The the weekty 
Gueteines) are varied at the discre- conference. 
tion of the director and include informal talks 
and lectures by him, papers by the members on 
assigned topics, reports by the members on the 
instruction they have witnessed ; in the case of 
members of the faculty, this report is confined 
to a statement of the facts observed, while in 
the case of colleagues in the seminary, the 
method of teaching may also be discussed and 
suggestions for improvement made. The exer- 
cises are not very formal, and questions and 
discussions may constitute an important feat- 
ure of the proceedings. The seminarists are 
also guided in the reading of pedagogic litera- 
ture. 

The aim is to shape the work of the seminary 
so as to cover the most important topics dur- 
ing the course of the year; such as the consti- 
tution of the higher school system, the curric- 
ula of the schools, the aim of the work as a 
whole and of its various parts, the methods of 
teaching in general, more detailed considera- 
tion of the teaching of those subjects which the 
members of the seminary expect to teach, the 
subject-matter of these latter subjects from the 
teacher’s stand-point, text-books and other aids 
and appliances, etc. 


20 Teaching of Mathematics in Prussia 


Immediately on entering upon his work, the 
seminarist is set at observing the instruction 
Observation Which is being imparted throughout 
ofinstruction. the institution. About twelve hours 
per week are given to these visits of observa- 
tion which at first range over all subjects and 
from the lowest to the highest class, in order 
that the seminarist may understand the scope 
of the work as a whole and the interrelation 
of its parts. Later, his visits are concentrated 
more upon classes in the subjects which he is 
preparing to teach, and finally upon a class 
which is soon to be put under his own instruc- 
tion fora time. As already mentioned, he pre- 
sents a report of his observations to the weekly 
conference, and his report receives the criti- 
cism of his colleagues and of the experienced 
teachers who may be present. 

The seminarist also attends the regular fac- 
ulty meetings, but he has no voice in the dis- 
cussions except when called upon to report 
concerning pupils under his charge. 

When the seminarist has thus visited a desig- 
nated class for a sufficient length of time to be- 
actual come familiar with the character and 
instruction. methods of the work being done, he 
is permitted to give the instruction himself 
under the direction and supervision of the per- 
manent teacher of the class. The entire re- 
sponsibility for the work done still rests upon 


The Teachers 21 


the latter, who is usually present during the 
hour, though less frequently as the candidate 
progresses satisfactorily. The director and 
other candidates also often look on. On occa- 
sion, the teachers present correct the candidate 
in the class-room, point out defects or suggest 
better methods, and he is supposed to take coun- 
sel with them privately as to his work. The 
other seminarists report on his teaching and 
criticise it freely at the weekly conference, and 
the teachers who saw him at work add such 
remarks as they may deem wise. The candi- 
date teaches only a few weeks in any one class, 
when he is assigned to similar work in another 
class, his assignments being at first to lower 
classes, and later to the higher classes. 

In the seminary year the candidate is entitled 
to no remuneration, though there are some 
small stipends. 

At the close of the year the director submits 
to the provincial school-board a full report on 
the work of the seminaries; if the work of any 
has been unsatisfactory, he is debarred from the 
opportunity of proceeding farther in the course 
of preparation for the profession of teaching, 
while the others are advanced by the board to 
the ¢rial year. 

This year (Probejahr) is usually passed in a 
different institution from that in which the sem- 
inary year was spent, and in it the candidatus 


22 Teaching of Matbematics tn Prussia 


probandus is given six to eight hours per week 
of instruction to do, and the classes are placed 
under his charge as their regular in- 
structor, though he still works under 
the supervision and guidance of his superiors. 
In some cases the exigencies of instruction make 
it necessary to assign him more instruction to 
give than that mentioned above. He is paid by 
the hour for what teaching he does. Prior to 
the institution of the seminary year (1890), the 
trial year included work of the character now 
done in the seminary year. 

We have thus seen that, apart from the time 
requisite to pass the extended and searching 
Summary of €Xamination pro facultate docendt, a 
preparation. minimum of five years of special prep- 
aration is required of everyone who would 
become eligible to appointment as teacher ina 
Prussian higher school. Three of these years — 
are devoted to preparation in subject-matter 
and two to learning the art of teaching; in the 
first of the latter, theoretic study of pedagogic 
problems and methods preponderates, while in 
the second the candidate tentatively begins the 
practice of independent teaching. There is 
small wonder that few of those who survive all 
of these tests prove later to be poor teachers, 
and that, on the whole, German teaching leads 
the world. 

Having passed through the seminary year 


The trial year. 


The Teachers 23 


and the trial year satisfactorily, the candidate 
is eligible to appointment as teacher. This may 
be either provisionally as assistant The ati 
teacher (wzssenschaftlicher Hiilfsleh- ointment. 
rer), Subject to dismissal upon three months’ 
notice, or definitely as instructor (Oderlehrer), 
removable only for serious offences and after 
formal trial.! 

In the interests of the work instructors may 
be 

a. Transferred to other positions of not lower 
rank and not less pay, with payment of moving 
expenses ; 

6. Placed temporarily upon the inactive list 
with prescribed pay; or 

c. Placed permanently upon the retired list 
(Ruhestand), with corresponding pension (see 
below). 

The Minister grants the title Professor to in- 
structors who have evidenced scientific or ped- 
agogic excellence,and upon the same Ranks and 
grounds professors may receive the ponares 
rank of Councillors of the Fourth Class (Kathe 
vierter Klasse). This gives them social equality 
with the Councillors of the Fourth Class in the 
other branches of the government, with men of 


1 In certain urgent cases instructors are suspended or forbidden 
to exercise their functions. They receive half pay while suspended, 
and the other half is also paid to them if upon trial they are found 
innocent. 


24 Teaching of Ahatbematics in Prussia 


high rank in’ other professions, who are Coun- 
cillors of the Fourth Class, such as University 
professors, and the like, and gives them social 
precedence of such as have not this distinction. 
The conferring of these titles has no effect upon 
the salary of the recipient. 

The appointed teacher takes a prescribed 
oath of office by which he becomes an official 

of the state. As such he must culti- 
. vate loyalty to the king and the 
realm, advance the interests of education and of 
his institution as far as he can, and in particular 
give instruction not to exceed the following 
number of hours per week: 

Director, 14-16. 

Instructor, 20-22. 

The “hours” are at most fifty minutes, and 
at the close of each hour there is a pause of at 
least ten minutes. The maximum number of 
hours may be required only if the classes are 
small and if the subject of instruction does not 
demand from the teacher a time-consuming 
correction of papers. 

The teacher also is to do without extra pay 
such emergency teaching as may be rendered 
necessary by the death of a teacher of the insti- 
tution and also to replace free of charge such 
of his colleagues as may be absent through ill- 
ness, through leave of absence on the ground 
of ill-health, through being called into military 


Duties. 


The Teachers 25 


service, or through jury duty. If ateacher ob- 
tains leave of absence for any cause or other 
than one of these mentioned, he must himself 
defray the cost of filling his place during his 
absence. If a teacher is obliged to travel for 
his health, to visit springs or baths, an appro- 
priation in addition to his regular salary may 
be granted him in view of the extra expense to 
which he is subject. 

The salary of an instructor is $648 per an- 
num (1 Mark = 24 cents) during the first three 
years, at the close of which time he re- 
ceives an increase of $72 per annum, 
and each three years thereafter a like increase 
is made until in twenty-four years the maximum 
salary of $1,224 is reached. One-half of the in- 
structors in the complete institutions (see be- 
low) and one-fourth of those in the incomplete 
institutions receive annually a fixed addition to 
their salary (feste Zulage) of $216. This addi- 
tion when once attained is permanent and is usu- 
ally granted new recipients in order of senior- 
ity as vacancies occur (by death or transfer to 
the inactive or retired list). Instructors also 
receive an appropriation for rent varying with 
the population of the town in which the insti- 
tution is located. This appropriation ranges 
from $216 for Berlin to $86 in the smallest vil- 
lages. This is the only item in which the in- 
comes of instructors vary according to the 


The salaries. 


26 Teaching of Mathematics in Prussia 


location of their institutions. Consequently 
the incomes of instructors will gradually in- 
crease in Berlin from $864 to $1,656, and in the 
smallest villages from $734 to $1,526, while 
other cities and towns have ranges intermedi- 
ate between these. 

After ten years of service instructors are en- 
titled to pension in case they are permanently 
Thepen-  Gisqualified for teaching by physical 
sions. _or mental weakness or by incapacity 
for the work. The amount of the pension de- 
pends upon the number of years which the 
teacher has served, and also upon the income 
which he is receiving at the time of retirement. 
The entire income of the instructor at the time 
(salary, rent allowance, and fixed addition, if he 
is the recipient thereof) is the base upon which 
the pension is computed. After ten years of 
service the pension is one-fourth of the base, 
and for each additional year of service one-six- 
tieth is added to this fraction, until after forty 
years of service the maximum pension of three- 
fourths of the base may be received. In count- 
ing years of service the seminary year, the 
trial year, and military service performed after 
the age of twenty are included. Instructors 
are not required to contribute to a pension 
fund. 

In Berlin the amount of pension would ac- 
cordingly range from about $234 at the end of 


The Teachers 29 


ten years’ service to $1,227 at the end of forty 
years of service. 

After forty years’ service instructors are 
privileged to retire upon pension even though 
not incapacitated, but this is rarely  The«jupie 
done. Inthe programme of one in- lee.’’ 
stitution a long list of instructors was pub- 
lished who have taught in that institution up 
to their “Jubilee” (fifty years of service). Each 
of the older institutions could doubtless furnish 
a more or less extended list of this kind. "The 
completion of fifty years of service is usually 
celebrated appropriately. The veteran “ Fubz- 
lar’ often receives letters and tokens of dis- 
tinction from the government, addresses in rec- 
ognition of his long and valued services from 
his colleagues, and tributes of respect from his 
pupils, present and past. 

The deference paid to age and experience 
in the German school-world is very marked. 
The responsible posts, the directorships, the 
portions of the work of instruction considered 
the more important or desirable, are usually 
allotted to elderly or old men. The younger 
men work with them in cheerful subordination 
and with genuine respect for the greater wisdom 
of greater experience. 

If we wish to consider the total professional 
income of the Prussian teacher, we must take 
the assurance of pension into account. Pensions 


28 Teaching of Mathematics tn Prussia 


can be purchased in life insurance companies 
in Berlin, by the payment of a fixed sum an- 
Total profess NUAally for a term of years, and the 
sionaline study of their rates may enable us to 
aeok form an estimate of the market value 
in cash of the assurance of pension which the 
Prussian teacher enjoys. The conditions on 
which the pensions are paid by the companies 
differ somewhat from those of the schools as 
outlined above, otherwise the rates of the com- 
panies might be taken as being the cash value 
of the assurance of pension. The following 
table is constructed from the prospectus of a 
strong company doing business of this sort. 

If payments are begun at the age of twenty- 
five, the annual payment of P dollars until the 
age A is attained would secure at the age A a 
lifelong pension equal to that which the Prus- 
sian teacher would receive if retired upon pen- 
sion at the age A. 


Awssagess 35°./40.5 45 9.50 ° 55 s0omuaoe 
Beat alelan S $366 $300 $243 $242 $197 $142 $94 
NOTES 


1. Different teachers retiring at the same age A might not re- 
ceive precisely the same amount of pension owing to difference in 
the ages at which they began service as teachers and the age at 
which they attained the fixed addition, The range of variation 
would not be very considerable ; in the above table an estimate of 
the average has been used. 

2. In making the table it was assumed that the fixed addition is 


The Teachers 29 


attained between the ages of forty-five and fifty years. The teach- 
er’s pension at the latter age would thus be quite considerably 
larger than at the former, which accounts for the very slight de- 
crease in the amount of the annual payment P, requisite for the 
purchase of the same pensions to take effect at these two ages. 


The conditions of assurance correspond quite 
closely with those to which the teachers’ pen- 
sions are subject, with one important exception 
—viz., the payment of pension by the company 
begins at a stipulated date, neither sooner nor 
later, and the state of health of the assured or 
his need of the pension has no influence what- 
ever on his receipt of it. The annual payment 
of $351 for a period of ten years assures defi- 
nitely that payment of pensions will be made 
from the age of thirty-five on, while the teacher 
receives pension at thirty-five only if perma- 
nently incapacitated for work. Likewise the 
annual payment to the company of $197, for 
instance, assures the payment of pension at the 
age of fifty-five, dut not before, no matter how 
urgently it may be needed. The teacher, on 
the other hand, has the continual guarantee that 
he will receive pension whenever he may need it. 

It is not easy to deduce from the table above 
the annual payment which should be made to 
secure precisely the same guarantees which the 
school pension system gives. A very rough 
approximation may perhaps be obtained by re- 
garding the average of all the rates—v7zz., $224— 


30 Teaching of Matbematics tn Prussia 


as expressing the amount of the mean annual 
payment sought. Assuming this as the average 
annual cash value of the pension guarantees, 
the incomes of the Prussian teachers when 
compared with those of non-pensioned teachers 
should be considered as ranging in gradual 
ascent, sure to each teacher, from $1,088 to 
$1,880. 

To compare these salaries with American 
salaries, the difference in the purchasing power 
Purchasing ©! Money in the two countries must 
bower in be taken into account. Conservative 
America. : 

estimates made by those who have 
lived for extended periods of time as settled 
residents in each country regard the purchas- 
ing power of money in Germany as about four. 
third times that in America. If this be correct, 
the German incomes would be equivalent in 
value under American circumstances to a range 
from $1,451 to $2,507 per annum. 

The American work in mathematics corre- 
sponding to that for which these salaries are 
paid in Germany is done in the grades below 
the high school (five years), in the high school 
(three years), and in the freshman year in col- 
lege, the younger teachers receiving the small- 
er salaries and doing the work of the earlier 
years. 

The above is a crude attempt to estimate the 
value of the pension guarantees, but by far the 


The Teachers 31 


most beneficial effects of the Prussian pension 
and salary system are not capable of inclusion 
inacash estimate. The Germans ap- Tranquitity 
preciate that the teacher can do his oe. 
best only in an atmosphere of financial and men- 
tal tranquillity. He must himself be continually 
growing, and if he is embarrassed by financial 
cares and harassed by struggles to improve 
his material position, his growth is retarded 
and the quality of his work inevitably deterio- 
rates. The teacher is spurred on to his highest 
achievements by devotion to his calling and by 
the inspiration of seeking, finding, and impart- 
ing truth, but not by the competition of the 
mart or by the goad of necessity. 

The educational system of Prussia recognizes 
this truth, and while insisting on high standards 
and severe tests at the outset, assures a tranquil 
career to those who have given evidence of 
their fitness. 

The German teacher works with a sense of se- 
curity ; security in his position without regard 
to the occurrences of politics or the whims of 
the powerful and the influential; security ina 
modest competency while at work; security in 
case of a “rainy day;” security in his profes- 
sion as a lifework, in the assurance that if, as 
a young man, he maps out for himself a pro- 
gramme of work, study, and research requiring 
decades for its completion, nothing but the flag. 


32 Teaching of Mathematics in Prussia 


ging of his own assiduity or the collapse of his 
physical or mental powers will prevent its suc- 
cessful execution; security, finally, that after a 
quiet life of patient, undistracted, fruitful toil 
in his noble vocation he will reach a well-cared 
for and honored old age, with all needed repose 
from his well-finished labors. 


V 
The Pupils 


Boys who apply for admission must have 
completed their ninth year, must be able to 
read German in both German and 
Latin characters, have a clear and 
legible handwriting both in German and Latin 
script, be able to write from dictation without 
bad orthographic errors, know a little biblical 
history, ard be familiar with the four fundamen- 
tal overations on whole numbers. 

The pupils are classified into nine classes, 
each with a course of one year. The following 
table gives the names of these classes, ctassification 
the abbreviations which we shall use  9"448¢ 
for them, and the average age of the boys in 
each near the close of the year. In forming 
this table, the writer examined statistics as to 
the average age of pupils in twenty-three insti- 
tutions. These institutions were selected so 
that all kinds were included and different loca- 
tions represented. The variations in the aver- 
age ages as between institutions were found to 

33 


Admission. 


34 Teaching of Mathematics in Prussia 


be so slight that these twenty-three institutions 
were regarded as giving a correct idea of the 
range of ages. The average age for each class 
is published by every institution. The average 
of these averages constitutes the average of 
the table. The highest and the lowest average 
found for each class is also given: 


Class. 


Oberprima. 
Unterprima 
Obersecunda. 
Untersecunda. 
Obertertia. 
Untertertia. 


Abbreviation...| IA. || IB. | ITA. | IIB. | LITA.) TITB.) TV. even ae 
Average age. ..| 19.4 | 18.4 | 17.4 | 16.5 | 15.4 | 14.3 | 13.2 | 1r.8 | 10.7 
Highest aver- 


age age...... 20:9 | 19,4 | 18.6 | 17.2 | 25.97| 15.5 [23.0 Mere gare 
Lowest aver- 
ape age... ss. 18.2) }.27.5 ] 16.3 | 25.8 24.5 1523.5) (era, 0 ek Sars 
Lowest age 
possible for 


any single 
pupil, about..] 17.9 | 16.9 | 15.9 | 14.9 | 13.9 | 12.9 | 11.9 | 10.9] 9.9 





The table shows that the variation in age 
among the pupils of any one class cannot be 
very great, and the appearance of the classes 
themselves confirms this. 

The class-names given to the pupils are 
formed by adding zer to the names of the 
classes: as Primaner, Sextaner. 

The teachers of each class, in conference, 
name the first doy or Primus of the class. The 
appointment is made on the basis of scholarship 


The Pupils 35 


and is considered a high honor. The Przmus 
acts as monitor, supervises the room in the ab- 
sence of the teacher, keeps the class record, 


and says prayer at the beginning and the end 
of the day’s session. 


VI 
The WMnstitutions 


The institutions are divided, according to the 
character of the work and the length of the 
course, into the following six classes 
‘(the customary abbreviations being 
given in parentheses): 


Classification 


A. Nine years’ course. 
Gymnasium (G), has both Latin and 
Greek. 
Realgymnasium (RG), has Latin but no 
Greek. . 
Oberrealschule (ORS), has neither Latin 
nor Greek. 


B. Six years’ course. 
Progymnasium (PG). 
Realprogymnasium (RPG). 
Realschule (RSch). 


The institutions under B do precisely the first 
six years’ work of the corresponding institu- 
tions with nine years’ course. 

The institutions of Class A are called “ Com- 
plete Institutions” (Vol/anstalten), and those of 

36 


The Unstitutions a7 


Class B “Incomplete Institutions” (Vichtvollan- 
stalten). 

The number of institutions of each kind in 
Prussia was in 1896 as follows: 


Pe IMIASIETE Pulte Ses '4)s/d) ee os e\ e's! 273 
PGA TEINASIOI 0.1. )r e's! o'e ol ob! e al 86 
Mierreaschulene yc e tea 24 
POMS GL senior aul) ig 45 
Pea LOD WV INIMASIONy 1121) cele 9 s'¢e 71 
| Cie EST Con bck Sk an Ue nS 73 

Ue hte aa weet te rcs ce a 572 


Another classification: 


Witiewatinand Crreeki.J..i.1. 5 5318 
With Latin but no Greek. ....157 
With neither Latin nor Greek. 97 


We have already stated that these institu- 
tions do not correspond closely to any kind of 
American school, and that hence the German 
names will be retained in speaking of them. 
The ideals, administration, and discipline are 
throughout strictly those of the school, but the 
curriculum extends somewhat beyond that of 
our best high-schools. In mathematics, the 
work done covers approximately the same 


38 Teaching of Mathematics in Prussia 


ground as our course to the close of the fresh- 
man year in college. 

In addition to their income from tuition fees 
(about $33 per annum per pupil),and from en- 
Financial dowments (usually not large), insti- 
sgt ie tutions may be supported by the state, 
by the city in which they are located, by private 
organizations, or by private individuals. The 
state makes good deficiencies in the budget of 
institutions not primarily supported by it, and 
all institutions, even those receiving no support 
from the state, are obliged to conform to the 
same curricula and are subject to the same in- 
spection and control by the educational authori- 
ties. 

The school-buildings are usually located in 
the interior of blocks, away from the noise and 
The build: bustle of the street. In Berlin they 
weer are as arule modern and well adapted 
to their purpose, those of the last few decades 
especially being models of school architecture. 
There is a strange deficiency in all the build- 
ings, even the newest, in two respects: only one 
blackboard, from five to eight feet in length, 
is to be found in each room, and the seating 
arrangements are not good, usually consist- 
ing of desk-benches holding from four to six 
pupils. When called to the board the older 
pupils pass before the others of the same row, 
while in the lower classes the boys often run 


Tbe Tnstitutions 39 


along back of the others on the benches on 
which they are seated. There isalwaysa large 
hall for physical culture (Zurnhalle, gymnasium 
in our sense of the word), fitted out with appa- 
ratus, and a yard for open-air exercise whose 
area (roughly estimated) is at the minimum 
about equal to that covered by the buildings 
and at the maximum is several times as great. 

Each institution has two good libraries, one 
for the pupils and the other for the teachers. 
Catalogues of the first are sometimes The equip- 
published indicating the classes to ment. 
which each book is suited. The second con- 
tains important scientific and pedagogic works, 
and both libraries are increased each year by 
purchase and gift, the titles of the new acquisi- 
tions being mentioned each year in the Pro- 
gramm. The physical and the chemical labora- 
tories and the museums of natural history 
seemed to be good, but the purpose of the 
writer’s visits did not call for careful examina- 
tion of them. Mathematical models do not ap- 
pear to be in much favor. 

A large, cheerful room, called the “ Confe- 
renzzimmer,’ is always set apart for the social 
and official use of the faculty. Along fhe tacuity- 
the side walls are ranged desks and ior. 
lockers for each teacher, and through the centre 
extend long tables amply provided with writ: 
ing material, including red ink. 


40 Teaching of Mathematics in Prussia 


This room is the working centre of the insti- 
tution. From here pulsates the life-current of | 
the work, and the influence of this simple, easily 
provided centre gives tone, vigor, and unity to 
the activities of the entire teaching body. Here 
each teacher meets his colleagues several times 
daily, and exchanges a few social phrases with 
them or arranges some detail of their common 
work; here each keeps his needed books and 
papers, refreshes himself, rests quite free from 
responsibility during the ten-minute pauses and 
works in quiet when he has a free hour; and 
here finally, the Director drops in frequently for 
a little chat or to speak about this, that, or the 
otherof the routineoftheday. Inall these ways 
the constant use of this room must contribute in 
no small degree to unity and efficiency of work. 

The sessions are held on all six week-days 
and the arrangement of hours in the day is 
fixed by each school in accordance 
with localneeds. The vacations vary 
slightly, but usually are about twelve weeks in 
length, divided as follows: 


Sessions. 


BLAStOT. i 2st. 24 iol os tle aR eee 16 days 
Woihitsuntigey 0 0 ae be le eee Wikee 
Summer (usually including July) 38 “ 
Michaelmas. (2 i ig te as one Oct 
ACO ISOS: p's \e's)e wtp hie ek et ea 1G ies 


Che Institutions AI 


In addition there are various single holidays, 
and in summer the session is often omitted or 
abridged on hot days. 

The year is divided into two semesters, the 
Michaelmas and the Easter; if the number of 
pupils is large enough, there are two full sets 
of classes, in one of which the year begins at 
Michaelmas, in the other at Easter. When this 
is the case, a pupil who fails to be promoted 
falls back only six months. When there is only 
one set of classes the year usually begins at 
Easter, though not invariably. 

An interval of ten minutes intervenes between 
the ringing of the bell which terminates one 
exercise and the ringing which is the 
signal for the beginning of the next. 
During each of these pauses the pupils are re- 
quired to leave the room and take exercise in 
the yard; if the weather is bad they remain in 
the corridors. A pupil remains in charge of 
the books and clothing left in the classroom, 
and in each corridor and in the yard a teacher 
exercises supervision, but otherwise all are free 
during the pause. It is a period of relaxation 
and refreshment for all. Both pupils and 
teachers bring a supply of sandwiches to be 
eaten inthe pauses. Noises are tolerated freely 
both in the corridors and on the playground; 
the pupils run about and shout, but I saw no 
concerted plays. Indeed, the American visitor 


The pause. 


42 Teaching of Mathematics in Prussia 


to Germany is soon struck by the fact that the 
children gambol about aimlessly and do not 
play games. As a few minutes are necessary 
for the pupils to return to their rooms after the 
bell has rung, the actual period of instruction of 
each “hour”’ is about forty-seven minutes. 


VII 
The Curricula 


The first plan of studies uniformly followed 
throughout Prussia was issued in 1837. Prior 
to that only the rules for final exam- First 
inations (Adzturzentenexamen), first is- curriculum. 
sued in 1812, and isolated ministerial orders, 
were binding upon the schools, which on the 
whole arranged their curricula independently. 
The class system was introduced throughout 
Prussia in 1820, and there have been during the 
present century four revisions of the course of 
instruction—v7zz., in 1837, 1856, 1882, and 1892. 

The last revision was preceded in the autumn 
of 1890 by a Conference called by the minister 
of education to discuss a number of  theconfer- 
questions submitted by him. The ence of 1890. 
forty-four members included representatives 
not only of the schools concerned, but also of 
the Universities, the Educational Administra- 
tion, the Church (both State Church and Roman 
Catholic), and the Army. Among the members 
who are well known in America were Paulsen, 
Helmholtz, Virchow, and Zeller. The delibera- 

43 


44 Teaching of Mathematics in Prussia 


tions of the conference extended over several 
weeks and were opened and closed with an ad- 
dress by the Emperor in person. The com- 
plete stenographic report of all the delibera- 
tions has since been published’ and will repay 
careful study. 

Among the important problems considered 
were the question of obligatory Greek (the spe- 
cific question at issue being further preroga- 
tives for the real-institutions), the extent of the 
Latin instruction, and the overburdening of the 
pupils. 

The first resulted in no additional preroga- 
tives for the real-institutions; the second, ina 
marked diminution of the number of hours of 
instruction devoted to Latin, an increase in the 
number of hours given to German, and the 
abolition of the Latin theme as an end in itself 
(Zielleistung); and the third resulted in a dimi- 
nution of the hours of instruction in all the 
schools (see curricula below), and in a sharp 
demand that home-work should play a very 


minor part (see methods, etc., below). 

It is with much regret that this mention of 
these instructive deliberations is permitted to 
suffice. The interested reader will find the 
published report very suggestive. 

Passing to the consideration of the curricula 


1 Verhandlungen tiber Fragen des hiheren Unterrichts, Berlin, 
1891, pp. 800. 


The Curricula 45 


themselves, we give first the detailed curricu- 
lum of the gymnasium as fixed by the plans 
of 1892. The abbreviations already fhe currice 
mentioned (p. 34) are used for the «lum of the 
classes, and the figures denote hours #77" 
of instruction per week throughout the school 
year. 


CURRICULUM OF THE GYMNASIUM. 





VE Ve v. IIIB. IIIA, |ITB. ITA.} IB. } IA.| Total. 

















Christian Religion...... ETE We ee 2 2 2 heads | 19 
ermancncdeiene. ss sass2 4 } x t 3 2 2 3 Suwa are 
(DEN a als Gee ee ee 8 8 7 7 7 6 6) (76) 62 
GG APL ltt Psee 6 6 6 Gore Go 36 
RONG Heerassydie'cs sys clasts sie 'e Satta ted 3 3 2 ZalesttenLo 
History and Geography.| 2 | 2 | 4 3 3 3 Shes: (i 20 
Mathematics): 6 3..:.. AM irae Wid. 3 3 4 Fide ehh icy 
Natural History........ 2 le) to 2 a ed SW aid pone 8 
Physics, Chemistry, and 
Mineralogy......... By HAE 2 2 EAP ie Bl lee fe: 
WISE Kero cle itecls ceases Deo el | ae ae 4 4 
PAWS is cde esse dues ce zaS 2 2 7 8 
aL otalin =). pe teapy « 25) 25 oorlego) 44,30 28 | 28 | 28 | 252 
NorESs. 


xz. German and Latin in VI. and V. are to be taught by the same person if 
possible. 

2. Three hours per week of physical culture are required of all classes. 

3. The portion of the table to the left of the heavy line constitutes the curric- 
ulum of the progymnasium. 

4. The instruction in the Christian Religion is given according to the tenets 
of the State Church (Protestant) or those of the Roman Catholic Church, or 
both, as circumstances may demand. Jewish pupils and others who cannot 
conscientiously attend either the Protestant or the Roman Catholic instruction 
are dispensed from the requirement in the Christian Religion, on satisfying the 
authorities that they are receiving an equivalent amount of instruction in tenets 
of their own religion. If the number of Jewish pupils is sufficient, instruction 
in the tenets of the Jewish religion is given in the school, in the amount re- 
quired by the curriculum, 


46 Teaching of Matbematics in Prussia 


The above plan indicates sufficiently clearly 
the way in which the hours in each subject are 
Synopsis of 1Stributed among the various classes. 
other In the other kinds of institutions and 
curricula. in the earlier curricula the distribu- 
tion is, in principle, the same, so that it will 
suffice to give for them the totals correspond- 
ing to those contained in the last column of the 
above table. (The unit is one hour of instruc-. 
tion each week throughout one year.) 


CURRICULA OF THE PRESENT CENTURY. 





Real- Oberreal- 











Gymnasium. gymnasium. schule. 

1837|1856| 1882] 1892|| 1859} 1882/1892) |1882]1892 

Christian Religion. ; oi) -uiespas0s 18 | 20 | 19 | 19 |; 20 | 19 | 19 |) 19 | 19 
Gernian ices esic ema le cele opie sere 22 | 20 | 21 | 26 || 29 | 27 | 28 || 30 | 34 
GeV EYIa bere rive: slats Bis oe's inte aoe tations 86.|'86 | 77.}'62 || 44) S45) 4st ei 
Greek Oo aasclo can motes ww atelepnarrs 42) 42e) Aowls36.Wl 2.) cee joa giete 
PROP T ag techs cre aves eve sie ferale, Clete oleae 12, | 17 | 2x] 19 || 34 | 34 | Sr uh) SOllay 
IBPISh. Geettciie ae eat wnat oars ae fae e pose Plate afl 20°] $20 Te Siete 
History and Geography......... 24 | 25 | 28 | 26 || 30 | 30 | 28 || 30 | 28 
Mathematics icity slesisis as sites oe 32 | 32 | 34 | 34 |! 47 | 44 | 421) 49 Woa7 
Natural History 6| &| 20] 8 || 18 | x2 | 22 || 13 | 22 
Physics fo ead a ieiata eg i 6} 8) z0*}) 8. | a2) een erie 
Chemistry 220 spic aise voto siale'sic's nie eat deel Aakoat rss 81 206 ap 9 | 11 
Writing lees seein cemeteries af Grane: 4 7 4 4 6 6 
LAWINH, |.) dalenis es pik ahaielatele's ie eats 6}° 61 6} S j)20) s8ahrenh reneieae 
eLotal aovcssiectaec oe coetlen 258 | 268 | 268 | 252 || 285 | 280 | 259 || 276] 258 











* Includes the elements of chemistry and mineralogy. 


The ministerial rescript offers the following 
regulations concerning the instruction in math- 
ematics in the gymnasium: 


The Curricula 47 


a. General aim of the instruction.—Facility of 
calculation with numerical quantities, and their 
application to the usual circumstances 4) cuericue 
of everyday life. Literal arithmetic to tum in math- 
the binomial theorem for positive in- = “4° 
tegral exponents and algebra to quadratics, both 
inclusive. Plane and solid geometry. Plane 
trigonometry. The idea of codrdinate and 
some of the fundamental properties of conic 
sections. In all of these subjects not simply 
an intelligent knowledge of the theorems is to 
be reached, but also skill and facility in their 
application. 

6. Scope of instruction.—The topics treated in 
the various classes are as follows: 

VI. Four hours per week.__Review of the fun- 
damental operations with whole numbers, both 
abstract and denominate. German weights, 
measures, and coins, with exercises in the deci- 
mal notation and the simplest calculations with 
decimals. 

V. Four hours per week.—Divisibility of num- 
bers. Common fractions. Simple exercises in 
proportion, to be solved by reduction to unity. 
German weights, measures, and coins as in VI. 

IV. Four hours per week— Arithmetic, two 
hours. Decimals, simple and compound pro- 
portions with integers and fractions (exercises 
from practical life). Plane geometry, two hours. 
The straight line, angles, triangles. 


48 Teaching of Mathematics in Prussia 


IIIB. Three hours per week.—Literal arith- 
metic, one hour. The fundamental operations 
with absolute numbers, restricted to the most 
necessary matter. In the exercises equations 
of the first degree are also to be used. Plane 
geometry,twohours. Parallelogram. First part 
of circle. | 

TILA. Three hours per week.—Algebra (first 
half-year, one hour; second half-year, two 
hours). Equations of the first degree with one 
and several unknowns, herewith exercises in 
fractions. Powers with positive integral ex- 
ponents. The most necessary things concern- 
ing radicals. lane geometry (first half-year, 
two hours; second half-year, one hour). Theo- 
rems concerning quality of areas of figures. 
Computation of the areas of rectilinear figures. 
Beginning of the theory of similarity. 

I1B. Four hours per week.—Equations, includ- 
ing simple quadratics with one unknown. 
Powers with negative and fractional exponents. 
Concept of logarithms. Exercises in compu- 
tations with logarithms (five place). Compu- 
tation of circumference and area of circle. 
Definition of trigonometric functions. Trigo- 
nometric computation of right and isosceles 
triangles. The simple bodies, with computa- 
tion of lengths of edges, surfaces, and volumes. 

IIA. Four hours per week—The theory of 
powers, roots, and logarithms, Equations, in- 


The Curricula 49 


cluding quadratics with several unknowns. 
Arithmetical and geometric progressions. 
Conclusion of the theory of similarity, golden 
section, something on harmonic points and 
pencils. Plane trigonometry, with exercises in 
the computation of triangles, quadrilaterals, 
and regular polygons. 

IB. Four hours per week.—Review (by means 
of exercises) of the algebraic work of the earlier 
classes. Compound interest, annuities, imagi- 
nary quantities. Completion of trigonometry 
(addition theorem). Solid geometry and mathe- 
matical geography on the sphere. 

IA. Four hours per week.—Binomial theorem 
for positive integral exponents. Conclusion of 
solid geometry. The notion of co-ordinates 
and some fundamental properties of conic sec- 
tions. 

The degree of thoroughness with which these 
various topics are handled may be judged from 
the texts used, the actual work seen in the class- 
room,and the examinations set at the close of 
the course. The specimens of the examination 
papers which we shall give later will perhaps 
enable the reader to form his own opinion as to 
the scope of the instruction. The text-books 
and the class instruction will also be discussed 
in another connection; it may be said here that 
all the available information seems to indicate 
that on the whole the German work is at least 


50 Teaching of Mathematics in Prussia 


equal in extent and thoroughness to that of thé 
better American schools. 

c. Methodic remarks.—The ministerial rescript 
adds a few directions under the title “ Method- 
ic remarks,” among which are the following: 

The teaching of arithmetic is to aim at secur- 
ity and facility in operations with numbers. 
That it may be in harmony with the following 
algebraic instruction and prepare for it, the re- 
views of the fundamental operations in Sexta, 
as well as the treatment of fractions in Quinta 
and Quarta, must be based upon mathematical 
form, and the handling of parentheses must 
likewise be continually practised. In fractions 
the pupil is to be taught to operate with frac- 
tional parts as concrete things. The instruction 
_in arithmetic as such stops in Quarta, but se- 
curity in computation is to be maintained by 
continued numerical exercises in the algebraic 
instruction of the following classes. 

Strict adherence to the work assigned to each 
year is an absolute requirement. As it is more 
difficult in mathematics than in other subjects 
to replace deficiencies in elementary attain- 
ments by private industry, and as experience 
has shown that the difficulty which this subject 
sometimes presents to the pupil in the upper 
classes is almost without exception due to 
deficiencies in the foundations, conscientious 
strictness in the promotion of pupils becomes 


The Curricula sr 


the more an urgent duty toward the pupils 
themselves. 

We pass to the mathematical curriculum of 
the Realgymnasium and of the Oberrealschule. 
The scope of the instruction in the Mathematics 
real gymnasium is as follows: is hii 

Algebra, including the proof of the schools. 
binomial theorem for arbitrary exponents and 
the solution of equations of the third degree. 
Plane geometry, including the theory of har- 
monic points and pencils, and points and axes 
of symmetry. Solid geometry and the funda- 
mental propositions of descriptive geometry. 
Plane and spherical trigonometry. Introduc- 
tion to the theory of maxima and minima. 
Plane analytic geometry. 

In addition to these, there are required in 
the Oberrealschule : 

The most important series of algebraic an- 
alysis. Equations of the fourth degree and the . 
approximate numerical solution of algebraic 
and transcendental equations may be taken up 
at the option of the instructor. 

In all these topics the work is to give prac- 
tice in the application of the theorems, as well 
as to lead to a mastery of the proofs themselves. 

The directions for the work in mathematics 
have been very sharply criticised by the math- 
ematicians in the schools affected. It is not 
necessary to reproduce these criticisms here or 


52 Teaching of Mathematics in Prussia 


to make comments of our own with the one 
exception of calling attention to the sad defi- 
Fie ciency in the “General Aim of the 
strictures. Tnstruction” which is set up to 
guide the teacher in his work. No unifying 
principle is offered for making the mathematical 
work a harmonious whole, no suggestion of 
treating this subject as a portion of general 
culture; nothing but the attainment of a speci- 
fied amount of mathematical technique is fixed 
as the aim of the work. 

The distribution among the various classes 
Distribution of Of the hours allotted to mathematics 
the hours. in the different curricula, appears in 
the following table: 


Gymnasium. 

VI. Vv. IV. IIIB. IIA. IIB, IIA. IB. IA. Total. 
1837....0. 4.4 (3 3. 3. 34. 4. ee 
1856...... 4°93 3 3 BS 4 SS 
1882... .06 AR BOB 3 4 A eee 
TROIS suid 4.4 (4 63 2 3° ©4014 % > Sage 

Realgymnasium. 
ISSO. 24 6:0 5 4 6 6 6 5 5 5 5 47 
1882.....- §S «4.5 5° 96 ).55.9 95) 5 345 eee 
18QI.eeeee 4 4 4 .§) $ °45.. 45 35 Spee 
Oberrealschule. 
TESS 55 ya 8 6 6 6 6 5 5 5 5 49 
I8OL. 250. 5 5 0726 5 5 5 5 were}: 


VIII 


The Tnstruction in Matbematics 


The arrangement of the classroom, the 
benches and the small blackboards, have already 
been mentioned. Wet sponges are  ctassroom 
used as erasers. Light (a/ways from cUatiasies 
the left of the pupils), heat, and ventilation were 
as arule adequate. The lower classes which I 
saw had from thirty to thirty-five pupils, while 
in the three upper classes the number ran from 
nine to twenty (the last number was exceeded 
in only two instances, Obersecunda with thirty 
pupils). The pupilsare all assembled when the 
teacher comes and when he enters they invari- 
ably rise and remain standing until he bids them 
to be seated. They assume the same attitude 
when he leaves the room at the close of the hour, 
and also whenever any other member of the 
faculty, or a visitor like myself, enters or leaves 
the room. The pupils are addressed as Du to 
the close of Obertertia, thereafter as Sze. 

The first thing which impressed me in the class- 
work, and that which remains finally the most 
prominent characteristic, was that the ¢eacher 

53 


54 Teaching of Mathematics in Prussia 


teaches. We does not “hear recitations;” he 
does not examine the pupils to see whether or 
Theteacher nOt they have learned some assigned 
poouetce: matter from a book; this custom 
seems happily quite a thing of the past here. 
At times he imparts new knowledge himself, es- 
pecially by way of definition and introductory 
work, but most frequently he leads the pupils 
on by skilful questions themselves to discover 
new truths. In the development of new propo- 
sitions the teacher guides the work, but the 
pupils suggest step by step what is to be done 
next. 

Home-work and the study of books are 
very minor features; by far the heaviest stress 
Stresson 15 laid on the class-exercise. Here; 
theclass- under the carefully planned instruc- 
exercise. = tion, under the direct influence of 
the personality of the teacher, the progress is 
to be made. Private work and the study of 
text-books have simply the purpose of fixing in 
mind or giving practice in that which is sup- 
posed already to have been learned. 

The teacher is the source of the pupil’s 
knowledge and the authority on which he 
builds. “ What does the book say?” is a 
question never raised in a German school; in 
all my visits, I heard no books referred to, 
except collections of exercises. 

Whether it is best to train pupils to such 


Tbe Mnstruction in Mathematics 55 


marked dependence on personal, oral guidance 
in the acquisition of knowledge is open to ques- 
tion. I am here and throughout simply stat- 
ing the facts as I found them without discussing 
their merits either when regarded alone or in 
comparison with other systems. An educa- 
tional system is far better judged by the results 
it accomplishes under fair and sufficient trial 
than by any quantity of theorizing about it. 
The results reached under the Prussian system 
will be discussed later. 

This is, however, the proper place to say that 
the Prussian system does produce most excel- 
lentteachers. Thethorough preparation which 
they are required to undergo both in the scien- 
tific and the practical pedagogic lines tells dis- 
tinctly in their work ofinstruction. There were 
many remarkably strong teachers among those 
whose classes I visited, and among them all 
there was only one that could be called un- 
questionably poor. 

If I were to describe the method of instruc- 
tion taken as a whole by a single phrase, I 
should say it is the “ Socratic ee 
method,” the method of skilful ques- Socratic 
tioning, of leading the class on tothe ™"*" 
desired goal by a series of questions, each 
usually fairly easy to answer in itself. Except 
in case of review-questions to refresh the mem- 
ory or to recall the material needed for the day’s 


56 Teaching of Mathematics in Prussia 


work, the questions have a clear didactic pur- 
pose and value, and generally give evidence of 
having been carefully planned. Every bit of 
the hour’s work is vitalized by the teacher; 
there is nota minute when his voice is not 
heard and there is also not a minute when his 
voice only is heard. 

I was especially impressed with the general 
custom of dividing the work into very simple 
steps, and of repeating each new fact established 
over and over until it seemed that it must be 
imbedded in the mind of the slowest, before 
going on to the proof of the next. This is very 
distinctly characteristic of the German class- 
room. The questions are very simple, often half 
suggesting the answer, but still leaving some- 
thing for the pupil to think out and add. One 
director, in praising his leading teacher of 
mathematics most highly, said: 

“He hammers away and simply makes the 
pupils follow the work; the secret is that he 
works for the slowest.” 

It may seem that this mode of procedure 
is to the disadvantage of the brighter pupils by 
holding them back to the pace set by the weaker 
portion of the class. Indeed, several German 
mathematicians have remarked to me upon the 
slowness of the progress and have recalled how 
irksome. this had sometimes proved to them as 
pupils. The problem of how to care for the 


The Mnstruction in Mathematics 57 


weak and the average pupil without holding 
back the talented pupil to his detriment seems 
to be as far from solution in Prussia as in 
America. If, however, the galaxy of mathema- 
ticians who have sprung from the benches of 
the German gymnasia be taken into considera- 
tion, the question may well be raised whether 
or not the retardation of the gifted pupils is in 
fact to their detriment. 

The answers are always given in complete 
sentences, and clear and distinct enunciation is 
insisted upon. Every lesson in math- 
ematics is thus more or less of a lesson 
in German. Considerable stress is also laid 
upon the oral solution of easy exercises. In 
review, quite complicated problems are thus 
proved. I heard, for example, boys about 
thirteen years old (Untertertianer) prove the 
Pythagorean theorem with no figure whatever 
before their eyes. Different pupils did not use 
exactly the same lettering for the figure. The 
teacher informed me that this was not a mere 
feat of memory, but that the pupils could fol- 
low the proof on an imagined figure and that 
they enjoyed this kind of work. I saw this in 
various forms in several classes, and the pupils 
enter into it with considerable zest. Simpler 
new theorems are also proved orally, in some 
cases with the figure on the board; in others 
the figure was constructed and all were allowed 


Oral work, 


58 . Teaching of Matbematics in Prussia 


to take a good look at it and fix the image in 
mind, when the figure was erased before the 
proof was begun. 

Work in concert is effected in the Diarium 
or exercise-book. Some exercise is dictated 
The exercise. Dy the teacher; the pupils work it 
book. simultaneously, one reading as he 
works; the same pupil reads only one or two 
steps, so that quite a number are called on before 
the exercise is finished. The reply which the 
pupil makes seems to be the only means which 
the teacher has of determining in how far each 
pupil has worked along with the others and 
understood the step taken. The exercise-books 
are not usually inspected either during or after 
the class-work. Sometimes the teacher or a 
pupil works on the board, the others working 
along on paper, or looking on and dictating in 
response to the questions of the teacher ; some- 
times the teacher works on paper with the 
class; sometimes he has the exercise in hand, 
already worked out. Sometimes the result 
found is discussed, reformulated by members 
of a class until a satisfactory form is reached, 
and then copied as a theorem for future use. 
The “Diarium method” could readily be 
adapted to work at the board by the whole 
class in concert. 

Whether working in the diarium or at the 
blackboard, the pupils are trained from the 


> 
f 


The Wnstruction tn Matbematics 59 


very beginning to read aloud distinctly what 
they write as they write 7t—to “ chalk and talk.” 
This habit might well be cultivated «cham ana 
in American pupils. One of the talks 
chief advantages of an oral explanation is that 
one sees the proof growing and taking shape, 
and comments can be made on each step as itis 
taken and its correctness and purpose satisfac- 
torily explained before the next step is made. 
In the same way, figures may be constructed 
line by line as required. Indeed, by far the 
best way to master a proof which is put before 
one in complete form is to write out the proof 
step by step on paper, constructing what fig- 
ures may be needed as the proof proceeds. If 
complete figures are prepared beforehand at 
the board, and the reasoning is written out in 
full before any oral explanation or discussion 
takes place, the possibilities of oral work are 
very imperfectly utilized. In an oral explana- 
tion the word of mouth is of chief moment; 
the writing is auxiliary and of the character of 
a record of what has been explained. 

The attainment of any degree of skill in the 
threefold activity of performing an operation, 
describing it orally, and recording it in symbols, 
requires systematic and persistent practice 
throughout the entire work in mathematics. 
The German boys in the lowest classes were as 
ready in writing and reading simultaneously 


60 Teaching of Mathematics in Prussta 


the operations they had to perform as were 
the boys in the higher classes in their more 
complicated work. This is so because the 
teachers insist on it from the very beginning, 
by example as well as precept. 

It might be well to encourage the pupils to 
practice at home explaining proofs aloud and 
writing them out simultaneously. In addition 
to its direct purpose, this practice would also 
contribute to fixing the spirit and methods of 
the proofs themselves more firmly in mind. It 
should therefore be confined strictly to proofs 
which have been fully explained in class and 
clearly understood, lest, otherwise, errors and 
erroneous conceptions should become more 
firmly rooted. 

The following lessons may illustrate some of 
the characteristics of the instruction that have 
Alessonin JuSt been described. The first, a les. - 
aigchr, son in algebra, is taken from my notes 
of a class visit, while the second, a lesson in 
geometry, is a model for the study of begin. 
ners, set up in a German work on the teaching 
of mathematics. 

The algebra lesson was given in Untertertia, 
the first year in which algebra is taught, the 
minimum age of the boys on entrance into this 
class being eleven years. 


1Reidt, Anleitung zum mathematischen Unterricht an hoheren 
Schulen, 1886, p. 31 et seq. 


The WTnstruction in Mathematics 61 


First, the following problem from the book 
of exercises in algebra! used in this class was 
taken up. 


ee 2 tee 
Neng)! hy 608 Ta 


6 st be 


All wrote the problem, one (John, say) read- 
ing aloud as he wrote and adding: 

“ We seek first the common denominator.” 

Teacher. “Yow do we do that? By a 
rule?” 

John. “No, by inspection.” 

Teacher. ‘“ Right. What is the common de- 
nominator ?”’ 

John. ‘“ Twenty-four.” 

Teacher. “Right. What do we do next, 
Henry?” 

Henry. “We multiply both members by 
twenty-four.” 

Teacher. ‘“ What is the result, William ?” 

William reads as all write, 


“t2r— 8x + 64 —4r + 34+ 24 = 264.” 


Teacher. “What do we do next, Karl?” 
Karl. “We unite the terms in the left mem- 
ber.” 


1Bardey, Methodisch geordnete Aufgabensammlung, mehr als 
6,000 Aufgaben enthaltend, 23te Auflage, 1897. 


62 Teaching of Mathematics in Prussta 


Teacher. ‘Give the result, Fritz.” 
All write as Fritz reads and writes, 


Pate 20Ane 


Teacher. ‘ What do we do next, Peter?” 
Peter. “We divide both sides by eleven.” 
Teacher. ‘What is the result?” 

Peter reads and all write, 


AE met Fb 


A problem just like this was worked simi- 
larly, and then, as this was one of the days on 
which home-work in mathematics is to be as- 
signed to this class, three problems of precisely 
the same nature were assigned by page and 
number from the book of exercises for home- 
work, viz.-} 

x a = 2th BE cig =, 
ashen 1 
2he — 3444+ 544-344 +1=-4%. 
ae 
ag 2 es 2b eet ea. 


This constituted the entire assignment for 
home-work. 


1 Bardey, Aufgabensammlung, p. 101, Nos. 73, 74, 75. 


The Mnstruction in Matbematics 63 


Next, the following problem was taken up, 
all writing and one reading as usual. 


2(7x — 10) — (50 — x) = 20. 


Teacher. ‘ What doesn’t please us here?” 

Various pupils raise hands and reply as called 
on by the teacher. 

“ The parentheses.” 

“ The known numbers on the left.” 

“ The fractions.” 

Teacher. ‘“ Which shall we remove first?” 

The pupils express different opinions. The 
teacher points out that the most practical order 
must be determined in each problem—“a mat- 
ter of feeling ”’—and then indicated that in this 
problem it would be easiest first to remove the 
fractions, then the parentheses, and then to 
rearrange and solve. All of this was carried 
through step by step as in the previous case. 

Then oral work was taken up. First, the 
expansions of (a + b)* and (a — b)* were re- 
hearsed both as formule and in words, and then 
a number of exercises were given, the teacher 
writing on the board and the pupils reading 
the results as called upon to doso. Very easy 
exercises were given at the beginning while 
those at the close were of the difficulty of the 
following : 


(44 — ay); (52?2+4)*; (+0; (4 — #4 — 2). 


64 Teaching of Mathematics in Prussia 


Then the formulz for (a + b)* and (a — b)8 
were deduced and repeated a number of times, 
and the hour came to a close. 

The geometry lesson is supposed to be 
given to the class Quarta, in the first year’s 
Alessonin Study of geometry, the minimum 
geometry. ace for admission to this class being 
eleven years. 

“The teacher draws a triangle ABC upon 
the board, and then questions the pupils some- 
what as follows, the pupils being called on 
singly by name in as lively alternation as pos- 
sible : 

How many angles has a triangle ? 

Name an angle of the triangle ABC. A sec- 
ond. A third. 

The teacher draws and defines an exterior 
angle, CAD. 

Who can draw another exterior angle? (Done 
repeatedly by various pupils.) 

How many exterior angles can be drawn at 
one vertex of the triangle? 

How many exterior angles can be drawn al- 
together? 

What are two exterior angles at the same 
vertex called with regard to each other? 

What therefore do we know as to the magni- 
tude of these two angles? 

How many exterior angles differing in size 
can a triangle have at most? 


The Mnstruction in Mathematics 65 


Why is it customary to speak of only one 
exterior angle at each vertex of a triangle ? 

In view of this custom, how many exterior 
angles would a triangle be said to have? 

For convenience, the letter at any vertex shall 
be used to denote the interior angle and the 
letter primed the exterior angle at that vertex. 
The notion of adjacent angles is supposed to 
have been explained earlier in the course. 

What are an interior angle of a triangle and 
its adjacent exterior angle called with respect 
to each other ? 

What theorem holds for two such angles? 

We wish now to compare the magnitude of 
an exterior angle with that of the two non-ad- 
jacent interior angles. For this purpose, we 
regard AB and AC as two non-parallel straight 
lines cut by a third, BA; the last produced 
forms the angle CAD or A’. 

In what position are A’ and B with respect 
to each other? Likewise A’ and C? 

Can therefore A’ = B, or A’ = C? 

To compare the magnitudes of the angles 
we draw AE parallel to BC and divide A’ into 
the angles CAE and EAD, which angles we call 
~« and y respectively. 

Since AE is parallel to BC, there is another 
angle in the figure equal to y; what is it? 

Why are y and B equal? Which are the 
parallels and which is the secant line ? 


66 Teaching of Mathematics in Prussia 


The teacher marks the equal angles with the 
same mark. 

Is there also another angle in the figure equal 
tox? What is it? The teacher also marks 
these equal angles with the same mark, different 
from that used with the previous pair. 

Why are these two angles equal? Whichtwo 
lines are now the parallels? Which the cutting 
line? 

Since y = B, and x= C, how large is x+y? 

To what is therefore the angle A’ equal? 

What theorem have we thus found ? 

The proof is now repeated synthetically, 
first with the same figure, then with a different 
exterior angle of the same triangle, or also with 
an entirely different triangle, something as 
follows: 

The teacher produces BC beyond C and 
asks: 

What do we assert concerning the exterior 
angle at C? 

What auxiliary line shall we draw to facilitate 
the proof? 

What pairs of angles are now equal? Why? 
(To avoid breaking the main course of thought, 
the parallels and the secant line are not now 
called for in detail.) 

What follows from the equality ? 

Karl, give the entire proof once more. (To 
hold the attention of the other pupils the teacher 


The Wnstruction tn Mathematics 67 


interrupts, if necessary, and calls upon others 
to give the reasons for statements made by 
Karl.) | 

In the next hour one or more repetitions of 
the proof are given in the same way until, if 
possible, all the pupils are able to present the 
proof in a connected manner.” 

“Many a beginner in teaching will regard a 
large part of the questions in the above speci- 
men as very superfluous and as use- _ pjections 
lessly squandering time, because he considered. 
thinks it may be taken for granted that the 
pupils know the answers. He will even fear that 
such seemingly trivial questions will not only 
not attract the pupil, but actually bore him and 
cripple his interest in the work through default 
of rapid progress to new matter. But if he 
makes the experiment of teaching in the manner 
- indicated, he will soon become convinced of his 
error. He will see that the great majority of 
the pupils are eager to participate in the dis- 
cussion, that they compete for the privilege .of 
answering the questions asked, and are rejoiced 
to know and to do something. He will also 
notice that the answer to these questions are 
by no means evident to some pupils, and he will 
be obliged to take special pains with these pu- 
pils. He will often enough be amazed at the 
colossal stupidity of some of the answers to his 
questions, but he will also be gratified to see 


68 Teaching of Mathematics in Prussia 


how, gradually, stupidity gives place to com- 
prehension, and the progress of the pupil, at 
first so slow, becomes more and more rapid, in 
consequence of the pupil’s own mental exer- 
tions. It will not be long before the stronger 
pupils at least will be able to make simple 
proofs without the previous assistance of the 
teacher.” 

The manner of the teachers was usually of a 
military sharpness, though not unkind; in many 
Manner of the CaSe€S it was mild, in a few genzal, and 
teachers. only exceptionally unkind or irri- 
table. The routine directions, especially, are 
given with the snap and precision of a military 
command and are met with an obedience equally 
military in its promptness and unanimity. This 
custom, together with the fact that the time 
spent by a teacher with the same class is meas- 
ured in years, permits the development of an in- 
formal code of directions (words and even gest- 
ures), by which the time occupied in giving 
and executing routine directions may be con- 
siderably reduced. There was often a sharp 
rattling fire of questions to which the an- 
swers came with corresponding promptness 
and precision, but at crucial points, where 
it seemed necessary for the pupil to collect 
his thoughts, ample time was allowed. The 
pupils were not only not hurried, but were 
openly encouraged to take time to think. 


Tbe anstruction in Mathematics 69 


“Take five minutes if you wish, oz/y make no 
mistake.”’ 

In commenting upon the work of pupils both 
the warmth of the praise for the good work 
and the severity of the censure for the poor 
work were more intense than they would have 
been under the same circumstances in America. 
It is safe to say that, despite the fact that many 
of the teachers of mathematics in our secondary 
schools are women, the emotional treatment of 
the instruction in Prussia is decidedly more de- 
monstrative than here. 

Concerning the written exercises nothing 
need be noted except the neatness of the 
papers. Isawanumberofsets,both ioe 
of final examinations and of class- of written 
work, and was present a few times it 
when the latter were returned with criticisms. 
All are written on uniform paper and any lack 
of neatness and mechanical accuracy is sharply 
criticised when the papers are returned. The 
result is that even the papers of the final exam- 
ination, when perhaps hurry and flurry might 
palliate careless writing, are to my American 
-eyes models of neatness. This standard can be 
reached only by unremitting insistence upon 
neatness from the very beginning and through- 
out the entire nine years. Indeed, I was told 
that the few papers which I singled out in one 
case as not up to the standard of the others, 


70 Teaching of Mathematics in Prussia 


were written by pupils who had entered the 
Gymnasium late and thus had not had the same 
drill as the others. 

The schedule of instruction is arranged so 
that any one class passes through the hands of 
as few different instructors in each subject as 
is practicable. During the nine years of the 
course of study the pupil has only one or 
two instructors in mathematics in the smaller 
institutions and from two to four in the larger 
institutions. Even in the latter, the aim is, if 
possible, to arrange the schedule of instruction 
so that the work in mathematics of the last 
three years shall be under one instructor, in 
order that the pupils may be carefully and sys- 
tematically prepared for the final examination. 

Whatever the number of teachers, the entire 
mathematical education of the boy from the 
Homoge. Clements of arithmetic to those of 
neity of analytic geometry takes place in one 
instruction. institution under one management, 
guided by the close supervision of the same 
director and under the tuition of men of the 
same scientific training, who are colleagues 
working in close contact, with opportunities 
for intimate interchange of ideas. Besides the 
director, the senior professor of mathematics 
gives more or less attention to the work of the 
younger instructors and thus contributes to 
uniformity of tone and spirit. 


The Wnstruction in Mathematics 71 


On the desk of each class there lies a large 
book, very durably bound, devoted to the rec- 
ord of the class’s work. One page is he ctass- 
allotted to each day’s work and one pete 
book serves a year. The page is provided 
with four columns, respectively for absentees, 
remarks, record of matter treated in the hour, 
and assignments. Horizontally, the page is 
ruled for each school-hour, so that at the close 
of the day a complete record of the day’s work 
in the class appears on the page. The home- 
work assigned being recorded, each instructor 
can see what the class is already required to 
prepare, and govern his own assignment ac- 
cordingly. Usually, the days on which home- 
work may be required at all are fixed for each 
subject, and I have at times seen this schedule 
posted on the wall of the classroom together 
with the hour-schedule for the class which is 
invariably there. The class-book is filled out 
beforehand as far as possible (dates, hours, sub- 
jects) by the Primus or first boy of the class. 
The other entries are made and signed by the 
teacher immediately at the close of the hour. 
In the column for remarks, anything not in the 
usual routine is entered, such as pupils excused, 
pupils misbehaving, reprimanded, punished, or 
doing very bad work. 

A few specimens will best show the nature 
of these entries. They are taken from some 


72 Teaching of Mathematics in Prussia 


class-books which I was kindly allowed to look 
through at leisure. They are all taken from 
Specimen  ifferent dates during one year, and 
entries. were made in one institution by va- 
rious teachers in different classes. 

A. was excused at 9.55. (The hour closed at 
ten.) 

B. occupied himself with outside matters dur- 
ing the hour. 

C. is not prepared. 

D. does not know the Homer verses by heart 
(the second time). 

D. does not know the Homer verses again 
(the third time). 

E. was reprimanded because of disturbance 
during the pause. 

F. is to blame for repeated lack of industry 
in Homer. The class was very noisy during 
the pause. 

G. knows nothing, and besides that answers 
pertly. 

H. is punished with one hour “ Karzer”’ (im- 
prisonment) because of persistent lying. 

J., on account of repeated unbecoming con- 
duct, receives the order to clean the sponge 
thoroughly daily. 

K. speaks with his neighbor during the 
written test and is punished with one hour 
“ Karger.’ 

The class-book is under the constant ispec- 


ae oe 


The fnstruction tn Mathematics 73 


tion of the ordinary, who thus keeps informed 
as to the work of the class, and the director is 
required to inspect all the class-books at least 
once each week. The pupils are also at liberty 
to examine it. 

All home-work is regarded as supplementing 
the work in the class and not asan integral part 
of the course. Its purpose is either 
the cultivation of neatness and order- 
liness in making clean copies of class-work, the 
memorizing of indispensable material, the fixing 
of what has already been learned, or the training 
to independent activity. Matter that has not 
been thoroughly explained in the class, so that 
the class as a whole understands it clearly, is 
never assigned to be studied privately by the 
unaided pupils. 

The quantity of the home-work is to be kept 
as small as possible. As maxima for the various 
classes, the following are officially suggested: | 


Stlmeveoely, LLIB, “LITA IIB: IIA; IB. TA. 
Meek 2 a 24 Sat 8:3 3 3 hours daily. 


Home-work. 


As already mentioned, the distribution of this 
time among the different instructors is arranged 
beforehand by the director or by the instruc- 
tors interested in conference, fixing the days on 
which each instructor may assign any home- 
work and the maximum of time which he is then 
at liberty torequire. The permanent record of 


74 Teaching of Mathematics in Prussia 


every assignment in the class-book enables col- 
leagues and superiors to keep track of each in- 
structor and see that he does not exceed his al- 
lotment. 

I learned the assignments in a few cases, and 
it seems, roughly speaking, that the amount 
Specimen Of home-work in mathematics re- 
allotment. quired per week in Prussia is consi- 
derably less than twice the amount required per 
day in the United States. As specimen of the 
allotment of time for private work, the follow- 
ing table may serve, giving the hours per week, 
as fixed for each subject and class in the Kgl. 
Realgymnasium, Berlin. 






































VI. Vv. IV. | IIIB.) IITA.) LIB, | TLAS) SIBSeEAS 
Christian Religion| 14 1} 14 13 1} It 1} 1} 14 
German. cocnnens 1 I I I I 1¢ 1; 2 2 
Latino nearee 3 3 4 2} 2t 2+ 2 2 2 
Krenchjcysc pb cntice aoe. Uh eee I 13 2 2 2 2 2 
Ron gliahh s+ sie:s ay’ t | 5 vival Same ae toe ees I 1} 1} 2 2 2 
Historyacesaes ce ° ° 4 4 4 I 1¢ 1¢ 2 
Geography ...... 4 3 3 & 3 4 | vise eee 
Mathematics, ... I I 14 2 2 2 24 2% 24 
Natural History. + 3 4 b 4 Men ee ee 
Physios Ske eal oe ee ome cae oP aie natalie ee lle or a I 1t 1 
Chemsintry 6) sik So hina tin ae oe pavtne OnE PRE Salah eile bree £ I I 
Total per week..| 7} 74 | roy | r1f | 12$ | 134 14 | 16 16$ 
Total per day....} 14 1} 18 133 | 23 2} 23} | 2§ 25 


Maximum per 
day suggested 
by Ministry...} 1 I 2 2 2¢ 2t 3 3 3 





In noticing the allotment to mathematics, it 
should not be forgotten that the scope of the 
work and the number of hours of instruction 


Tbe Unstruction in Mathematics 75 


in mathematics in the Realgymnasium are 
considerably in excess of what is done in the 
Gymnasium. 

The weekly amount is divided up into one, 
two, or three daily portions for each subject, 
and the days on which these allotments may be 
used are specified in such a manner that the 
aggregate daily assignment for private work 
is not far from the average on any one day. 
There are no “study hours” for pupils at 
school. While there, their time is occupied 
entirely with class-work. 

Ministerial rescripts of December 24, 1833, 
and August 16, 1860, require that 
some text-book be used in mathe- 
matics. The decision as to which book or books 


Text-books. 


1 The following is a free translation of these rescripts. 

(1833) ‘‘ The Ministry has had occasion to note that in many 
Gymnasia the instruction jn mathematics is conducted without a 
definite text in the hands of the pupils. In mathematics, if any- 
where, a brief text, suited to the needs of each class, is indispensa- 
ble. . . . In order to counteract these and other evils which 
have hitherto been more or less marked in the mathematical in- 
struction in the Gymnasia by reason of the lack of a definite text- 
book, the Ministry wishes herewith to fix that from Easter of next 
year a definite text-book is to be used in the various classes of all 
Gymunasia, that this text-book is to be in the hands of the pupils, 
and that no further attention is to be paid to any objections to 
this regulation which the teachers may raise. 

(1860) ‘‘The large number of text-books in mathematics and 
physics which are in use is an evil of considerable import. It is 
therefore very desirable that the use of those which have not proved 


76 Teaching of Mathematics in Prussia 


are to be used is made by the local authori- 
ties, but their choice is limited to such works 
as have been approved as sufficient for the 
purpose by the Ministry or the Provincial 
Board. Many books are used in one institution 
only. The presumption might be that in these 
cases the authors are teachers in the institu- 
tions using the book, but this is by no means 
always the case. To counteract the tendency 
to multiply text-books the authorities are be- 
coming more slow in placing new books on the 
approved list. It is now required that institu- 
tions with which the author is not connected 
shall have expressed their desire and purpose 
to use the book as text before it is placed on 
the approved list. Still, with all this discour- 
agement, the German teachers are writing new 
text-books in mathematics by the score every 
year. 

The text-book adopted is in many cases re- 
garded as named rather to comply formally 
with the regulation than for the purpose of 
actually using the book. This statement is 


strong books be still further discontinued and that they be replaced 
by more suitable works; nevertheless, the educational administra- 
tion will refrain now, as heretofore, from any direct constraint in 
this matter. The books adopted are often used too little, still, in 
attempting to enforce more use of the books, the danger would be 
incurred of hampering the more important effectiveness of the free 
individuality of the teacher.” 


The Tnstruction in Mathematics fir 


based both on information which I obtained 
from German teachers and on my own obser- 
vation. In all my visits I saw no books used in 
the classes (except collections of exercises and 
tables of logarithms), nor did I hear any allu- 
sion to the text in the work of the hour or in the 
assignment of home-work. One very excellent 
professor informed me, upon inquiry, that 
“Mehler ” was indeed officially in use, but that 
it was sometimes not alluded to in the class- 
work for months at a time. 

Whatever may be the extent of the actual 
use of the text-book, the German teachers are 
unanimously agreed on one point— purpose of 
the study of any particular topic in _‘ the text. 
the text-book must always follow the develop- 
ment of that subject in the class. The chief 
functions of the text-book are considered to be: 

first, to avoid the loss of time of instruction 
involved in the pupil’s copying the teacher’s 
explanations into a book as they are given. 

Second, to give those pupils who have not 
thoroughly understood the class presentation 
(and whose notes would probably also be 
faulty), a faultless presentation to which they 
refer and review the class-work, and clear up 
any points which had remained obscure. 

The books in use are accordingly written in 
a more or less brief form, as a skeleton, rather 
than as a complete body of instruction. Those 


78 Teaching of Mathematics in Prussia 


books which do give detailed treatment usu- 
ally profess to do so out of consideration of 
the needs of such as may wish to use the book 
for private study, unaided by a teacher. Of 
course, opinions vary widely among the teachers 
as to the degree of detail with which a book 
intended for class use should be written. Those 
actually in use differ much in this respect, and 
many teachers publish, for the use of their own 
institutions primarily, supplementary and full- 
er treatments of particular chapters, topics, or 
class allotments than the concise text gives. 


IX 
The Examinations 


One of the results of the Conference of 1890, 
as embodied in the new curricula of 1892, is an 
attempt to diminish somewhat the The exam- 
tasks of the pupils. Among other __ mations. 

‘things, the examinations have been made lighter. 

Annual oral examinations for promotion are 
held at the discretion of the Director and in his 
presence. So far as I could learn, these exami- 
nations are held on one or two subjects only in 
each class, and class-work alone determines the 
verdict in other subjects. 

The first formal examination comes at the end 
of Untersecunda and is known as the Adschluss- 
prifung. It covers the work of Untersecunda, 
and is conducted by a commission consisting 
of a Royal Commissioner, the Director, and 
the teachers giving instruction in Untersecunda. 
The questions are prepared by the teachers of 
the respective subjects and approved by the 
Director. The written examination in mathe- 
matics occupies four hours. In other respects, 
this examination is conducted like the final ex- 

79 


80 Teaching of Matbematics in Prussia 


amination, which will be described next. The 
certificate that the “ Adbschlussprifung” has 
been passed entitles the holder to a diminution 
of one year in the length of time of his mili- 
tary service (z.¢., he serves one year instead 
of two). 

The final examination, known as “ Rezfeprii- 
fung,’' or also as “ Abiturientenexamen,” is con- 
Thefinat | ducted by a Royal Commission, con- 
examination. sisting of a Royal Commissioner (who 
is appointed by the Provincial School-board, 
and is usually that one of its members who has 
the special supervision of the school in question), 
the Director, and the teachers giving instruction 
in Oberprima. Three months in advance the 
pupils give written notice to the Director of 
their desire to take the examination. The list of 
those applying is discussed in a conference be- 
tween the Director and the teachers in Oder- 
prima. The predicate to be assigned to each for 
his class-work is fixed (those in use are: Very 
Good; Good; Satisfactory; Unsatisfactory). 
By unanimous vote, those whose class-work 
has been unsatisfactory in all subjects may be 
excluded from the examination. 

The examination is both written and oral. 


1In describing this examination, we give simply a sketch of its 
characteristic features, without mentioning the exceptions and 
alternatives which may arise and which are duly provided for in 
the regulations. 


The Eraminations Sr 


The subjects of the written examination and 
the time allowed in addition to the  qhe written 
time mecaed to dictate the papers set® exemination. 
are as follows: 

German Theme, five hours; Mathematics, five 
hours; Translation from German into Latin, two 
hours; Translation from Greek and French into 
German, each three hours. 

Lexicons may be used, and in mathematics a 
table of logarithms. 

The mathematical paper contains four exer- 
cises, one each from Plane Geometry, Solid 
Geometry, Algebra, and Trigonometry. The 
papers set are afterward published in the Pro- 
gramm, and specimens will be given when the 
latter is described. 

In each of the subjects for examination— 
Latin, Greek, French, German, and Mathemat- 
ics—three examination papers are prepared by 
the teacher of that subject in Oderprima, and 
these papers, if approved by the Director, are 
sent to the Schulkollegium along with a list of 
the candidates and a sketch of their school- 
course. From each set of three the commis- 
sioner selects one to be used for the examina- 
tion and returns it in a sealed envelope to be 
opened at the time of the examination. He 
may reject all and prepare the paper to be used 
himself, if he sees fit. 

The papers are to make no requirements 


82 Teaching of Mathematics tn Prussia 


whatever which exceed in kind or difficulty 
Independent the work of Prima, yet they must 
work differ sufficiently from the class-work 
required. to demand independent thinking; in 
mathematics, in particular, the problems must 
all be so-called “ originals.” 

If the papers written by the pupils are such 
that the Royal Commissioner doubts whether 
the pupils have really done independent work 
in the examination, he has the power to require 
a new examination, the paper being set by him- 
self. The writing of the papers is supervised 
by the teachers of the class by turns. 

The papers written by the pupils are first 
criticised by the teacher of the subject in ques- 
tion; all errors, even though very slight, are 
carefully corrected in red ink, and at the close 
the paper as a whole is characterized in a few 
words, and a predicate assigned by the teacher; 
the predicate for the class-work is also men. 
tioned. 

A few specimens will illustrate the character- 
izations of the mathematical papers; these gen- 
eral remarks are in addition to a specific predi- 
cate for each problem: 

“The paper of A. is correct throughout and 
skilfully written, and also quite extensive. Very 
good. His class-work in papers of similar kind 
was good. 

“B. has indeed made two gross errors, still 


The Eraminations 83 


his paper may be called satisfactory in view of 
the ideas developed in it. His class-work in 
work of the same sort was satzsfactory. 

“C.’s paper is not free from obscurities and 
contains some mistakes, it is true, but it can still 
unhesitatingly be called satisfactory. His class- 
work has hitherto zot been satisfactory. 

“D.’s paper contains an oversight in the first 
problem, otherwise it is highly to be praised. 
Good. Class-work, good.” 

Next the papers are circulated for inspection 
among all the teachers of the Commission, who 
thereafter hold a conference with the Director 
and decide which pupils are to be recom- 
mended for exemption from oral examination. 
The corrected papers are then sent to the 
Royal Commissioner, who may alter the predi- 
cates. 

The oral examination must be held within the 
last six weeks of the school year at a date fixed 
by the Royal Commissioner who pre- The oral ex 
pigesseuctrcin. Iie determines the se- © s#aation. 
quence of subjects for this examination and the 
amount of time to be given toeach. All the 
teachers of the institution are required to at- 
tend the examination, from which are debarred 
those candidates whose written papers have 
all been “unsatisfactory,” and excused those 
whose written papers and class-work in all the 
required subjects have all been at least “ satis- 


84 Teaching of Mathematics in Prussta 


factory.” Partial exemption from the oral ex- 
amination may also be granted, and according- 
ly only those pupils appear in the examination 
whose work has been unsatisfactory in part, but 
not entirely so. At most ten may be examined 
in one day. 

The questions in each subject are asked by 
the teacher of that subject in Oberprima and 
by the Royal Commissioner. In the course of 
the examination, the predicates to be given to 
each pupil in each subject are fixed by the 
Commission, upon motion of the teacher of the 
subject in question, and, in conclusion, the Com- 
mission holds a consultation under the presi- 
dency of the Royal Commissioner in which the 
final outcome of the whole examination is deter- 
mined. The pupil passes normally if none of 
his predicates in the various subjects (formed 
by combining the predicates for the class-work 
and the examination) are “unsatisfactory.” 
With certain restrictions, the predicate “un- 
satisfactory ” in one subject may be counter- 
balanced by the predicate at least “good” in 
another. This consultation is not a mere me- 
chanical collation of predicates, but when neces- 
sary the merits of individual cases are delib- 
erated upon as such. All the proceedings in 
connection with the examinations are fully re- 
corded. The pupils who have passed receive 
a certificate to that effect, containing all the 


Ube Lraminations 85 


predicates and signed by each member of the 
Commission. 

The diploma of a gymnasium confers vari- 
ous privileges upon its holder, such as admis- 
sion to University study, to techni- priviteges of 
cal schools, to various examinations the graduate. 
for admission to certain military schools and 
the higher branches of the Government’s civil 
service, and is also accepted in lieu of some of 
the earlier examinations for officers in the army 
or the navy. Promotion to each of the four up- 
per classes opens to the pupil the door into some 
occupations which had been closed to him be- 
fore. Many of the privileges attained in the 
gymnasium may also be attained in the Real- 
gymnasium and in the Oberrealschule, though 
some of the most important may be attained in 
the gymnasium only. 


Xx 


The Programm 


The Director of each institution publishes 
annually. what is called the Programm, some- 
The scientific What analogous to our annual cata- 
papes logue. Itis usually in quarto form, 
substantially, plainly, and inexpensively gotten 
up. The contents fall into two main parts: A 
scientific paper (Wissenschaftliche Abhandlung) 
and the school-report. The scientific paper 
may treat an advanced topic, beyond the cur- 
riculum of the school, topics from the school- 


curriculum, or questions of pedagogic method. ~ 


It is not obligatory upon the incomplete insti- 
tutions to publish this paper. In 1896 there 
appeared 670 such papers in all Germany, the 
number of institutions being 993. 

The second part usually contains: 

I. The curriculum, in a table like that which 
we have used above. 
The schodl: IJ. The allotment of the hours of 
report. instruction to the various teachers, 
in a table which shows very clearly and readily 
the work of each teacher, and of each class. 

86 


The Programm 87 


III. The Class-work.—Abstracts of the work 
of each class in each subject, with specification 
of number of hours per week and name of 
teacher. The abstracts in other subjects are 
given with about the same degree of detail as 
those in mathematics, of which we give the fol- 
lowing specimens for the four upper classes of 
two institutions taken at random. 


Humboldt Gymnasium, Berlin, 1896. 


IA. Mathematics, four hours. Voss. The no- 
tion of co-ordinates; something about conic sec- 
tions; maxima and minima; series; computa- 
tion of the known functions. A paper to be 
prepared at home each month. 

IB. Mathematics, four hours. Voss. Com- 
pound interest; extension of trigonometry ; 
imaginaries ; solid geometry ; mathematical 
geography. Each month a home paper. 

IJA. Four hours. Schnodt. Trigonometry; 
completion of the theory of similarity; con- 
struction of algebraic expressions; detailed 
review of powers, roots, and logarithms; 
equations (several unknowns) which can be re- 
duced to quadratics ; arithmetical and geomet- 
ric series; reviews in trigonometry. Home 
paper every four weeks. 

IIB. Four hours. Schnodt. Extension of 
the theorems on similarity; trigonometric 


88 Teaching of {Mathematics tn Prussia 


functions in the right triangle; fractional and 
negative powers; logarithms; computation of 
the circumference and area of the circle; for- 
mule for the surface and the volume of the 
simplest bodies; equations of the second de- 
gree. Written papers every two weeks. 


Friedrichs Gymnasium, Berlin, 1896. 


IA. Four hours. Fischer, I. The number e- 


and the circle of numbers connected with it; 
supplement to solid geometry in connection 
with spherical trigonometry ; the notion of co- 
ordinates and the fundamental properties of the 
conic sections. 

IB. Four hours. Fischer, I. Imaginaries; 
combinations in application to arithmetical 
series of higher orders and to the binomial 
theorem; completion of trigonometry; solid 
geometry. 

IIA. Four hours. Fischer, I. Powers, roots, 
logarithms; arithmetical and geometric series 
of the first order with compound interest and 
annuities; conclusion of plane geometry; com. 
putation of the circle; plane trigonometry. 

IIB. Four hours. Schulze. Equations, in- 
cluding simple quadratics with one unknown; 
definition of powers with negative and frac- 
tional exponents; notion of logarithms; prac- 
tice in the use of logarithms; computation of 


Tbe Programm 89 


the circumference and the area of the circle; 
the trigonometric functions; computation of 
right and oblique triangles; the simple bodies, 
with the computation of their edges, surfaces, 
and volumes. 

Besides these are given the subjects assigned 
for themes in the various classes and the ques- 
tions set in the written examination for gradua- 
tion. These questions are scrutinized by the 
teachers in other institutions, and in mathemat- 
ics at least form a fertile source of fresh ma- 
terial in the line of exercises for class use. 

We give now a few specimens of the final 
examination-papers in mathematics in institu- 
tions of the various types. 


Friedrichs Gymnasium, Berlin, Michaelmas, 1895. 


weet. Lhe fiye roots’ are to: be deter: 
mined both algebraically and trigonometrically. 

2. A loan of 500,000 marks is to be ya sation- 
repaid by annual payments of 70,000 _ papers in 
marks. How many years will be re- "thematic: 
quired and how much will be the last payment? 
(Rate of interest not specified.) 

3. In a triangle, there are given one side, 
c = 748m., the difference of the adjacent angles, 
d = 11° 29.4’, and the ratio of their sines as 4/3. 
The triangle is to be solved. 

4. A segment of a material hollow sphere of 


9o Teaching of Matbematics in Prussia 


radius r = 533cm. has the weight 1.5 grm. per 
qcem. of its surface. If now this~basin will 
barely swim in water (sinking to the rim), what 
is the depth of the basin and how large is the 
radius of the circular rim ? 


Humboldt Gymnasium, Berlin, Michaelmas, 
1895. 


1. Given a circle and an ellipse of equal area 
with major axis a. A rectangle is to be in- 
scribed in the ellipse equal in area to the square 
inscribed in the circle. 

2. Given a circle and two tangents to it. 
They intercept upon a third tangent the length 
a. To find the angle between the third tangent 
and one of the others. To be worked first in 
theory and then computed for the following 
numerical values: r= 6.082, a = 35.025, and 
the angle between the two given tangents 
Oa oh 

3. A body of weight q falling from height h 
with initial velocity c penetrates s meters into 
the ground. How large is the resistance of the 
ground? To becomputed also numerically for 
h = 347m., C= 7m., S = 0,65m., and =e 

4. About a given spherical segment a cone of 
minimum volume is to be described. 


Tbe Programm Ol 


Luisenstadtisches Realgymnasium, Berlin, Easter, 
1895. 


1. What conic section is represented by the 
equation (rectangular co-ordinates), 


4x? — 4ryt+y? — 64V 4 — 87/442 Oe 
2. Solve the system 
KU = xKE=10; e+y—uU—2=4; e409 +y2 +2? = 130. 


3. In the geographical latitude of Berlin, 
what time lapses on the longest day until the 
middle point of the sun is 6° below the horizon? 

4. Given the radius r and the altitude h of a 
right cone, to determine the cylinder of maxi- 
mum volume which can be inscribed in it. 


Friederich Werder’ sche Oberrealschule, Berlin, 
Easter, 1895. 


I, From a triangle with sides a, b, c, a rec- 
tangle is to be cut such that the cylinder formed 
by bringing a pair of opposite edges together 
by bending a rectangle shall be as large as 
possible. 

2. The sinking sun shines into aroom through 
a circular opening in the front wall,and throws 
an elliptic spot of light on the side wall. From 
what point in the rear wall does this patch of 


92 Teaching of Mathematics in Prussia 


light appear to be circular? Construct this 
point, if the floor-plan of the room, the position 
of the circular opening, and the direction of the 
sun’s rays are given. 

3. By what equation is the radius of a sphere 
determined on which a triangle formed by three 
arcs of great circles has sides a, 2a, 2a, and the 
angle A between the equal sides? 

4. Investigate the curve 


4x? — Oxy +4lye+ 360% — 52v+10I =0, 


and compute the area enclosed by it. 


Eighth Realschule, Berlin, Easter, 1896 (Aé- 
schlussprifung—te., at end of Untersecunda). 


1. A wooden sphere sinks in water to one- 
half its height and in alcohol to 7/12 its height. 
What is the specific gravity of alcohol? 

2. On the side AB of a given quadrilateral 
ABCD arectangle is to be constructed equal to 
the sum of the three squares on the other three 
sides. 

3. In an arithmetical series, the sum of the 
second and the eighth term is 22, and the prod- 
uct of the third and the sixth term is 91. What 
are the first term and the difference of the 
series ? 

IV. List of Text-books Used.—Sometimes these 
are given in connection with the class-work 


The Programm 93 


(Pensen); sometimes no information on this point 
is given. The list is often in the form of a table 
making clear to the eye at once what books are 
used in each subject, in each class, in what 
classes each book is used, and the price of the 
book. The use of text-books in mathematics 
has already been discussed. 

V. Orders of the superior boards, so far as they 
are of general interest. 

As specimens we give the following, taken 
from the Programm of the Luisen Gymnasium, 
Berlin, 1896. When not otherwise specified, 
the orders are from the Royal Provincial 
Schulkollegium. 


1895. 

March 18. Physician’s certificate required with appli- 
cation for excuse from physical train- 
ing. 

neat. The 80th birthday of Prince Bismarck is to 
be a holiday. 
April 3. Arrangement of vacation courses in natural 
sciences and archeology. 

Dr. Fritz Bosch assigned to the institution as 

cand. prob. 

May 2. Days on which the institution has to flag. 

araneO. Approval of the rules of the boating di- 
vision, 

June Io. Tickets are received for five pupils and one 

teacher for the Luther Celebration in the 
New Market. 

ea To The work of Lindner on the war of 1870-71 is 
recommended, 


94 
August 2. 
ak ES « 
A ae 
et uirert & 
See Lae 


October 17. 


November 1. 


ce 2. 
é¢ 8, 
“6 19. 
“6 26. 


December 10. 


6e¢ ve 


«¢ 19. 


Teaching of Mathematics in Prussia 


Danger of firearms in the hands of pupils. 

Approval of the leave of absence of Professor 
Dr. Weber. 

The institution is to participate in the Sedan 
celebration in the Lustgarten. 

Freiherr v. Mirbach sends from the Civil 
Cabinet of Her Majesty the Empress an 
invitation to the consecration of the Em- 
peror William Memorial Church. 

To illuminate on September 2d (Sedan day). 


Permission to the institution to participate 


in the consecration ceremonies (October 
21st) of the Emperor Frederick Memorial 
Church. 

The vacations for 1896 are fixed. 

The introduction of the new French school- 
book by Ploetz Cares is allowed. 

Permission to add a seventh hour of Latin 
in Prima and in lieu thereof change one 
hour of physical culture into open-air 
games. 

Notification that Professor Ewald, Director 
of the Art School, will inspect the instruc- 
tion in drawing. 

The Stenographic Association is allowed the 
use of a room. 

Programme for the celebration of the 25th 
anniversary of the German Empire, on 
January 18, 1896. Addresses, songs, dec- 
lamations. 

Professor Dr. Gemss and Professor Dr. 
Weber receive the rank of Councillors of 
the Fourth Class. 

The Minister presents the work of Breysig: 
Brandenburg sche Finanzen. 


The Programm 95 


1896. 

January 17. From the administration of the privy purse 
(Schatullenverwaltung) of His Majesty 
the Emperor the institution receives as 
present the picture “/Vations of Europe,” 
by His Majesty the Emperor, with auto- 
graphic signature. 

af $e Three copies of Lindner’s book, the war of 
1870-71, are sent for presentation to good 
pupils; also three copies of the address of 
General von Mischke at the unveiling of the 
Emperor Frederick monument near Worth. 


fy Leave of absence granted to Professor Dr. 
Weber. 
ae a To illuminate on January 18th. 


ss 28. The workof Réchling and Knotel, Der alte 
Fritz, is commended. 

February 6. From Easter on, the Jewish religious instruc- 
tion is to be combined with that of another 
institution. 

27.  Wall-charts recommended. 


The above are about half of those published 
in this Programm as of general interest, and il- 
lustrate well how thorough and detailed is the 
supervision by the Schulkollegium. 

VI. Chronicles of the Institution.—This is a 
concise sketch of the life of the institution. 
Festal days celebrated are briefly described; 
occasionally an address by the Director is pub- 
lished in full; excursions made are enumerated; 
omissions of instruction for any reason specified 
in detail; changes in the Faculty are recorded ; 


96 Teaching of Mathematics in Prussia 


if new members are appointed a full account of 
their previous career is given; death or serious 
cases of illness among the Faculty or pupils, as 
well as any distinctions or honors that may 
have come to any of the Faculty, are suitably 
noticed; visits of inspection by the Provincial 
School Councillors (members of the Schudlkolle- 
gium) are described in detail, with words of 
thanks for the advice and encouragement re- 
ceived; in short, any deviation from the ordi- 
nary routine of the school is here made the 
subject of concise record, so that on one or 
two pages a clear picture of the school-life is 
painted. 

VII. Statistics—1. Summary of the attend- 
ance, by religion and residence. 

2. Table of the attendance in the various 
classes, gain or loss during the year, and the 
average ages. 

3. List of the graduates, giving for each, name, 
place, and date of birth, religion, years in the 
institution, years in Prima, occupation of father, 
and prospective occupation of the graduate. 

VIII. Additions to libraries, laboratories, and 
museums during the year. These are enumer- 
ated in detail, and all gifts, however slight, are 
mentioned in connection with the name of the 
giver and words of thanks. 

IX. Stipends and funds for the support of pu- 
pils. A report of receipts and expenditures, 


The Programm 97 


X. Notices to Pupils and Parents.—Information 
regarding the rules and administration of the 
school. Under this head almost all of the 
Programms in Prussia in 1896 brought in full 
a long, cautionary letter from the Minister of 
Education concerning the danger to pupils in 
handling firearms which was called forth by 
the accidental shooting of two pupils in this 
manner somewhere in Prussia. 

The information contained in the Programm 
is almost invariably grouped under the above 
heads, but their order is sometimes varied. 


XI 


The iReformschule 


In what has preceded, the work of the Prus- 
sian higher-school system has been described 
in so far as is necessary in order to understand 
the character of the work in mathematics done 
in these schools, but before closing this descrip- 
tion a few words may be added concerning 
two topics of some interest in this connection, 
the Reformschule and the higher education of 
women. 

In several institutions known as Reform. 
schulen the experiment has recently been inau- 
Character gurated of building curricula, in- 
and purpose. tended to be equivalent to the three 
now current, upon a common substructure up 
to the close of Untertertza, and of not separating 
the gymnasial and the realgymnasial course 
before Untersecunda. These institutions aim 
primarily to defer the decision between Greek 
or no Greek, Latin or no Latin, to a later period 
than the present curricula permit. As matters 
stand at present, the parents of the boy must 
decide which of the three classes of institutions 

98 


The Reformscbule 99 


he is to enter when he is only nine years of age, 
and when once the boy has made any progress 
in the work of any one class of institution he 
can transfer to another class only with consid- 
erable disadvantage. In the Reformschule, the 
choice between Latin and no Latin need not be 
made until the boy is at least twelve years old, 
and the choice between Greek or no Greek is 
then deferred two years further. To accom- 
plish this and still give an equivalent amount 
of Latin and of Greek it is necessary to give 
eight hours a week each to Latin and Greek 
during the last four years of the gymnasial 
course, and ten hours a week to Latin during 
the two years preceding these. The total 
amount of time given to Latin and Greek 
under this plan is not so great as that given 
under the present curricula, but it is believed 
by the advocates of the plan that the work 
done under the two plans will be equivalent, 
because the pupil should be able to accomplish 
more per hour in the latter part of his course 
than in the earlier years. To counterbalance 
the Latin and Greek thrown forward, French 
is thrown back into the earlier years, and more 
hours are allotted to it than under the present 
plans. In how far these and other expectations 
of the friends of the new movement will be 
realized remains to be seen. The ministry 
gives these institutions free scope for the trial 


100. = Weacbing of Mathematics in Prussia 


of the experiment, and the outcome is awaited 
with much interest. 

The first institution to begin the experiment 
was at Frankfurt a/M., where the work was 
begun in 1892; since then one institution each 
in Hanover, Breslau, and Berlin have in turn 
taken up the plan. As these institutions had 
previously been working on the usual curricula, 
the transformation of the institution into a Re- 
formschule-must be made gradually with the 
progress of the classes, those which had begun 
work under the standard curriculum having 
to be carried through according to the same. 
The process of transformation consequently re- 
quires nine years for its completion, and the 
first Reformschule, that at Frankfurt a/M., will 
graduate its first class as a Reformschule in 1901, 
and the other institutions several years later 
still, As no institution is as yet working under 
the new curriculum solely, it is too early to 
speak definitely of the outcome, and we close 
with the detailed curriculum of the Lezbnig 
Reformschule in Hanover, and a comparative 
summary table of the curricula at Hanover, 
Frankfurt, and the current plans of 1892. 


The Reformscbule IOI 


,CURRICULUM OF THE REFORMSCHULE AT HANOVER. 





Middle 


Substructure. era eure 


WE Vi IV. TTB TLEAS 


SSeS (Ce ey | 






































Chrrstian Religion... 5... ...,.5-5 3 2 2 2 2 
ERMIR ES as <a eiea idea esas. 5 4 4 3 3 
LEAL 5 4 SAAR Ste Aces eee Cite xe 10 10 
ROME doen cbc seins svcicceaessys 6 6 6 a 3 
retisiyciisine at ccaysiet- «cles we ele dees at +e 
Pe eh ngs siginiciw ¢ sivie cia. ny bis Ae AG Ae 
History and Geography ......... 2 2 5 4 4 
DVIATGOMIATCS 2 cc tales veces ems es ts 5 5 5 4 4 
PURTOTAL PAIStOSY...0 sk nee e esses 2 2 2 2 2 
SS By ae Ae Se She 
Chemistry and Mineralogy...... = oe : re 
WVIETUNIS PERI OA sors aiciuls isiic's eas. 3 ole. 2 2 2 ‘ie ate 
ror ese satere x sis icislels wveiaaasiecen 8 ai 2 2 2 2 
PRGtalterss sid sie ote ee ace te's 25 25 28 30 30 
Gymnasial Realgymnasial | 4 
Superstructure. | @ Superstructure. | $ 
H 
+ Oo 
Sls ie | ed is aS 
— i _ Ll _ _ it — 
Christian Religion,........... 2p 21 22 3G Shed Ca Gin ek eo 
EXTEIAT Es rats sis elt icisie sis sje'e's ans» 3 3 3 Sy ae 3 3 3 Saf (3% 
RARE can uidias oo 9't0 es Kas ole’ Soe 8 lS soo eS Boles he Se lees £0 
a cers es yrs vas voi a al ine ev a cs ae) 
HATS verte zc ors.s = =» rate ota ets Scie Y Aa Merete ie eats, ANA Oxia Ace ealeasierS 
Oe celle ee ea RatapOcwe Bera: fae. Moen at Preat Pi enmtres 
History and Geography...... Sime seie see to 2o Bil) Syd ey ie eet, 
CS eee yal pugs Be oie deh ite sad ove ha | Sa ad (be Je te a a, 
PVRPMEMIGUOLY see cccacces esi] s+ | S| ss | =» | TO Spelt sat eet pO 
oS eee Beles Sy 2. tes a 1S Sri Oh Pesos eon 
Chemistry and Mineralogy... Meet eek tte. boss Sto abe ein. G 
PI RIOR a cs se 0 Sept ipcad tite | Rasen er (3 age a i leat ecole, © ll» 
OO So} er hanes lease 8 PUAN irda ets: 
PRSERINMes's Paxert costae as 3r | 3z | 32 | 32 | 264 32 | 32 | 32 | 32 | 266 








ee 


102 Teaching of Matbematics tn Prussia 


COMPARATIVE SUMMARY OF CURRICULA. 


























Gymnasium. Realgymnasium. 
: ro : os 
o 5 o 5 
BP 1892. : Z 1892. 
i oC 
x cs x cs 
Christian Religion..... a ahs 19 19 19 19 19 
BORD Wyn ae a tukieehceeee 31 31 26 3r 31 
Latin o.\4. 2 Seo aioe o's cletealarats 52 52 62 40 40 
French ‘ E iigahia's 30 30 19 36 38 
Fingusnd, Aas acess. s AS nm 18 18 
POM ck ids has sb hoe ed nen: 32 32 36 ee >» 
History and Geography..... 28 24 26 29 27 
Mathematics 5. 05. C72 saan ce 38 37 34 43 42 
Natural History............ 10 10 8 10 10 
POP UCE Pes cas a eae Ee ay 8 8 Io 12 9 
Chemistry and Mineralogy. . as 3 be 6 6 
AV Siting «iss ces egieoraceeeee 6 6 4 6 6 
Drawing 5 nee fees sae tre 8 8 8 16 16 
MOA Joie Paces en deere 264 257 252 266 262 259 








XIT 
The higher Loducation of Women 


But little has as yet been done toward pro- 
viding for women similar educational facilities 
to those for men which we have just No gymnasia 
been describing. Nofull-fledged gym- fer women. 
nasia for women exist, and the proposal to 
found one in Breslau was vetoed by the Minis- 
ter during my stayin Berlin. In the latter city 
an arrangement has, however, been effected 
_whereby the “Courses of the Gym- What 
nasium’” (Gymnastalkurse) may be is done. 
taken by women under the instruction of vari- 
ous strong professors from some of the Berlin 
gymnasia. 

The courses are given in the afternoon when 
the instructors are free from their regular du- 
ties, and opportunity is given to those women 
who have completed courses equivalent to the 
curriculum of a Gymnasium to pass the regular 
final examination before some one of the Royal 
Commissions conducting the examinations of 
the Berlin institutions. The women who pass 

103. 


104 Ucaching of Mathematics in Prussia 


the examination successfully are admitted to 
University lectures under certain conditions. 

This work for women is under the direction 
of Miss Helene Lange, whose strong plea for 
the privilege of higher education for women is 
known to American readers in a translation 
published under the title “ Higher Education 
of Women in Europe.”! 

Through the courtesy of Miss Lange I was 
permitted to attend several classes in mathe- 
The quality Matics,and found the young women 
ofthe work. doing excellent work. As might be 
expected, they were distinctly more mature than 
the boys doing the same grade of work, and 
were decidedly more earnest and serious in 
their work; the social conditions in Germany 
are still such that a woman who seeks a higher 
education is regarded as a pronounced “ blue- 
stocking,” and this state of affairs deters all but 
women of fixed purpose and strong character 
from taking up these courses. With all the 
social traditions and prejudices discountenanc- 
ing this form of education for women and just 
as strongly urging it upon men, and even mak- 
ing it absolutely prerequisite to nearly all of 
the most desired careers, it is not surprising 
that, while boys of all degrees of talent flock to 
the gymnasia, only women of marked strength 


1Lange, Higher Education of Women in Europe, D. Appleton 
& Co., 1890. 


The thigber Education of Women 105 


are brave enough to carry through the corre- 
sponding work. 

Miss Lange has pointed out in the book men- 
tioned above that Germany is behind the other 
European nations (not to mention progress 
America) in the paucity of the facil- being made. 
ities for the acquisition of a higher education 
which it offers to women, but progress is being 
made, and year by year the privileges of women 
in this regard are being increased steadily, 
even though slowly. 


XIIT 


Comparison between German and American 
Work 


We are now ina position to consider in detail 
the facts upon which was based the assertion 
made at the beginning of this report—vzz., that 
in the corresponding nine years we Americans 
accomplish no more in mathematics than do the 
Prussians, and that we use up seven-fourths as 
large a fraction of the time of instruction in doing 
tt as do the Prussians. 

As the American basis of comparison the 
Chicago schools (Grades, High-Schools, and 
Basis of the University of Chicago) were 
comparison taken, in the belief that they are on 
the whole as typical of the better class of Ameri- 
can schools as any concerning which facts could 
readily be obtained, and that their work is at 
least up to the average of that done in insti- 
tutions of similar character throughout the 
country. 

The nine years in the Chicago system which 
correspond to the nine years of the gymnasial 
course are readily seen. 

106 


German and American Work 107 


At the age of nine the boy enters the gymna- 
sium, and here at theageofnine he g, nos 
enters the fourth grade and has had American 
something of fractions, decimals, and Sere 
denominate numbers in addition to what the 
German boy has had. 

He continues arithmetic during the next five 
years to the close of the eighth grade, but a 
part of the time allotted to mathematics in the 
latter grade is devoted to an introduction to 
algebra. 

In the first three years of the High-School, 
algebra through quadratics, plane and solid © 
geometry are taken up; during the fourth year 
of the High-School no time is given to mathe- 
matics, and consequently the first year in 
college constitutes the ninth year of mathemat- 
ical work; in this year plane trigonometry and 
college algebra are given twelve weeks each. 

The nine years of American school-work 
which are thus brought into comparison with 
the nine years of the German gymnasium 
are the five years next preceding the High- 
School, the first three years of the High-School, 
and the Freshman year in college. In the last 
year the American has one year’s advantage 
over the German as to age and mental develop- 
ment, but he has also the disadvantage of tak- 
ing up his work in mathematics after having 
suspended it for a year. 


108 Teaching of Mathematics in Prussia 


If the German curriculum as given above 
be compared with the ground usually covered 
Workdone in the nine corresponding years in 
compared. America, it will appear that the topics 
taken up are in substantial agreement, with the 
exception of the conic sections, something of 
which is contained in the German curriculum 
but not in the work of the American years in 
question. 

The text-books used and the exercises in 
the classroom both indicate that the Ger- 
man treatment of the topics is, on the whole, 
at least equivalent in scope and thoroughness 
to that in America. The test which the reader 
can most readily apply in this respect is to 
scrutinize the examination-papers which the 
German boys pass at the close of their work. 
The specimens given above may be regarded 
as sufficiently characteristic. 

Of course the German pupils have been gre- 
pared to pass such papers, even though the 
questions are all supposed to call for inde- 
pendent thinking at the time of the exami- 
nation, but perhaps the chief object of final 
examinations of this character is to test and 
exhibit the extent and thoroughness of the 
preparation of the pupils. There is little ques- 
tion that the German pupils would be able 
to pass the corresponding examinations given 
here. 


German and American Work 109 


The following table taken in connection with 
the preceding description of the Prussian gym- 
nasium, will substantiate the assertion Time ratios 
which was made at the outset. It compared. 
should be noted first that the comparison is 
made between fractions of the total time of in- 
struction and not between number of hours of 
instruction, for if four hours’ instruction out of 
eighteen weekly be given to mathematics, more 
work should be accomplished per week than if 
four hours out of thirty are given. The smaller 
number of hours per week instruction implies 
more time for private or home-work, and math- 
ematics is sure to secure its full share of this 
time. In fact, it is very possible that the aver- 
age American pupil gives to mathematics more 
than its proportionate fraction of his private 
work. In Germany, the allotment of timeto the 
various subjects, the record of assignments for 
home-work, and the methods and traditions of 
instruction all combine to make it improbable 
that more than the allotted portion of home- 
work accrues to mathematics. 

The table will be understood without further 
explanation. Inthe statement of the fraction 
of the total time of instruction given to mathe- 
matics the denominator is the total amount of 
instruction per week, and the numerator is the 
amount given to mathematics. The unit used 
is in some cases the minute, in others the hour, 





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“SOILVWAHLVW OL NHAID NOILONULSNI dO ANIL AO ATaAVL 


oy see eae 


German and American Work III 


and in the University the Course (of sixty hours). 
It will be apparent from the fraction which unit 
has been used. 

For purposes of comparison the table con- 
tains also the data for the Harvard School of 
Chicago, which gives under one administration 
the first eight of the nine years in question. 


XIV 


Conclusion 


Having now set forth the facts upon whose 
warrant it was asserted at the outset that a 
great disparity exists between the portions of 
time consumed in Germany and in America in 
doing work in mathematics which is in the main 
equivalent, it remains only to consider the in- 
fluences which seem to contribute to this differ- 
ence. The discussion of some of these merely 
emphasizes the fact that these influences seem 
to be well adapted to strengthen the work of 
Germans in Germany; it by no means follows 
that the same measures or similar measure 
would strengthen the work of Americans in 
American surroundings. The chief purpose of 
this report has been to place the facts before 
the reader in sufficient detail to permit him to 
form his own conclusions from the German 
system. At the same time, it may be permissi- 
ble to express, in the sequel, some of the opin- 
ions which have been formed after personal 
contact with both systems. 

Ii2 


Conclusion 0G, 


First, a few things which do zoft account for 
the disparity. It is not to be attributed to 
greater natural talents or to superior the dispar. 
mathematical ability in the average ity: how not 
German. It did not appear to me ee 
that the raw material for teachers or pupils 
was better than in America. In fact, as a na- 
tion, the United States has already given indi- 
cations of a marked mathematical dent ; to lead 
the nations in inventions, to be noted the world 
over as quick-witted thinkers (skilled in draw- 
ing conclusions, making inferences), to have 
as a national characteristic the extent to which 
individuals grapple with and solve practical 
problems as they present themselves (inde- 
pendent thinking, “individual initiative’), re- 
quires the same talents and aptitudes, the same 
forms of mental activity which lead to success 
if suitably applied to mathematical problems. 
Not every mathematician would be a good in- 
ventor, because the problems which the latter 
has to solve are much more complex than those 
of mathematics, but it can hardly be doubted 
that every successful inventor would have 
attained equally good results if he had, with 
proper guidance, turned his energies into 
mathematical channels. 

Nor is the disparity to be attributed to the 
fact that the German higher schools, which are 
not free and open to all as are the American 


114 Teaching of MMatbematics in Prussia 


schools, have to some extent “ picked pupils.” 
While it is possible that the German regulations 
may exclude a few pupils whose family antece- 
dents and home surroundings are not conducive 
to excellence in school-work, yet all the Ameri- 
can teachers who have been consulted (includ- 
ing several of wide experience) agree in the 
opinion that the mathematical work here would 
not be strengthened if the same regulations 
were put into effect (vzz.. a tuition fee of $30 to 
$40 charged). Some of the best work is often 
done by pupils whom a tuition fee would debar 
from the course, and the average of the ability 
and industry of the classes would not be raised 
if those were excluded who could not or would 
not pay a moderate tuition fee. 

The causes of the excellence of the Prussian 
work in mathematics may be classed 
under three heads: 

1. The Central Legislation and Supervision. 

2. The Preparation and Status of the Teachers. 

3. The Methods of Instruction. 

Of these, each is in a large measure a conse- 
quence of the preceding. Well-prepared teach- 
ers are likely to use good methods of instruc- 
tion and to evolve better ones. Thorough 
preparation of teachers is sure to be required 
when the legislative and executive authority is 
vested in experienced educators of the highest 
order. We shall consider the three heads in 


The causes. 


Conclusion 115 


detail immediately ; to summarize, it might 
almost suffice to say that the strength of the 
Prussian system may be said to be due to the 
fact that for two-thirds of a century the work 
has been centred in a single source of authority 
for the entire kingdom, whose directions have 
been carried out systematically and uniformly 
under the best guidance and supervision the 
nation could afford. 

I. Central Legislation and Supervision. — We 
have seen that the German school system is 
thoroughly organized on the principle of cen- 
tral authority. The teacher is not amenable to 
the local community, directly or indirectly. 
If in any quarter there is dissatisfaction with 
the work of a teacher, the only recourse is to 
bring the matter to the notice of his superiors 
in the educational work. Similarly, within the 
work, each is amenable only to his superiors, 
ending with the Minister, who is amenable only 
to the King. 

Each officer, from the Minister to the young- 
est teacher, is thus free to carry out his share 
of the work as may seem to him Expert gov- 
pedagogically best, with the assur. ¢rment. 
ance that only his superiors in the work, who 
have its success equally at heart with himself 
and who presumably are better qualified to 
determine its measures and policy, will sit in 
judgment upon his work. 


116 Teaching of Mathematics in Prussia 


There can be little doubt that this system 
secures the greatest efficiency in the work of 
the German people. A thoroughly military 
nation, trained for generations to respect for 
authority, accustomed from infancy to subordi- 
nation and prompt obedience to superiors, 
dominated in all business and social relations 
by the same spirit, the Germans have naturally 
developed a school system fitting best into 
their peculiar national temperament and social 
conditions. 

It is uncertain whether or not the same sys- 
tem would work equally well under other cir- 
Other cumstances. In particular, it is hard 
conditions. to determine what degree of centrali- 
zation is best in the United States. Despite 
the many considerable advantages of central- 
ized administration, some of the States (e.g., 
Massachusetts) which leave the administration 
of school affairs almost or quite wholly in the 
hands of the local community, have school sys- 
tems among the best in the country. 

The excellence of the Prussian system cer- 
tainly emphasizes one desideratum—vzz., that 
A desid- whatever may be the extent of the 
ers territory subject to one authority, 
whether a single village or an entire state, the 
actual authority, legislative as well as adminis- 
trative, be vested in expertenced educators. 

This is the practical way of looking at the 


Conclusion 117 


matter and is quite in consonance with Ameri- 
can business methods. So long as ultimate au- 
thority in educational matters is exercised by 
men who are not practical educators, who are 
in fact theorists in education, able at most to 
give some time spared from other business to 
speculations on educational questions, so long 
as non-educators have the right to say to the 
practical educators who are doing the actual 
work, “You are in our employ; we legislate, 
you have but to carry out our enactments,” for 
just so long will our educational system stand 
in the same danger of falling short of the best 
results which might be achieved that would 
arise if a body of teachers, be they never so 
earnest and well-meaning, were to try to man- 
age a great business enterprise in their leisure 
hours. 

Besides the general trend of centralization, 
a few direct results should be mentioned : 

1. Uniformity of Curricula.—This is not the 
place to discuss the desirability of uniform cur- 
ricula in all the institutions of the same character 
throughout a state or country. Modern habits 
of life, however, bring institutions into touch 
with one another in many ways and make a con- 
siderable amount of inter-institutional migration 
on the part of the pupils inevitable. The work 
of the pupils who have made a change, as well 
as that of the classes into which they enter, will 


118 Teaching of Mathematics in Prussia 


be more or less retarded if a readjustment of 
curriculum is necessary. Under the Prussian 
system, the transfer of pupils from one in- 
stitution to another causes the minimum of 
disturbance in the work of either pupils or 
classes. Practical uniformity of curricula is 
achieved and maintained better under central 
authority than in any other way. 

2. The Preparation of the Curricula.— The 
task of getting up or modifying the curricula 
for an entire country is a serious one and may 
well engage the time and attention of the great- 
est educators in the country, as in the Berlin 
conference mentioned above. It is not unwar- 
ranted to believe that stronger curricula will 
be set up in this way than when the curricula 
are prepared by local talent. Where there is 
no central authority, advisory bodies, like our 
Bureau of Education and various voluntary 
national and State associations, may make rec- 
ommendations, but their effectiveness is but 
slight in comparison with the enactments of a 
legislative authority, andthe mode of their prep- 
aration must be influenced by the fact that not 
a decision but a plea is being prepared, to which 
perhaps little attention will be paid by many of 
those who ought to profit by it most. 

3. Superviston.— The desirability of compe- 
tent supervision of the work of instruction is 
beyond question, and it is nowhere so highly 


Conclusion 11g 


developed and well organized as under a strong’ 
central system. When uniform curricula have 
been prepared by a central office, careful super- 
vision of the work naturally follows. 

4. American Conditions—There are in force 
in our country to-day only a few ¢ypes of cur- 
ricula regarded as representing dif- 
ferent educational principles or pur- 
poses, but there are unnumbered diferent curric- 
ula modelled upon one or the other of these 
types—curricula whose differences among them- 
selves are admittedly of no great educational 
significance, but which nevertheless require re- 
adjustment of work and almost inevitably en- 
tail loss upon pupils who are obliged to change 
from one institution to another. Considerable 
as this loss is, it is trifling in comparison with 
that which is due to the fact that during the 
nine years which correspond to the German 
gymnasium every one of our pupils swst make 
two changes of institution—from the Grades to 
the High-School, and from the High-School to 
the college. It is unfortunate that with us the 
work of these nine years is carried on in three 
different types of institutions, under different 
management, with different methods and aims, 
and with teachers differing radically in the 
character of their preparation. The subject of 
mathematics is very sensitive to these disturb- 
ances and disadvantages, and without doubt 


A serious loss. 


120. 3©Teacbing of Matbematics in Prussia 


suffers considerable through them. It is easier 
to point out the loss than to specify the remedy, 
but the ill effects of the break between the 
Grades and the High-School would be reduced 
and the mathematical work in particular would 
be much strengthened if the following recom- 
mendation of the Committee on College En- 
trance Requirements of the National Educa- 
tional Association! were put into effect: 


“TV. RESOLVED: That we favor a unified six- 
year high-school course of study, beginning with 
the seventh grade.” 


An admirable list of reasons in support of this 
resolution is given in the report. In the two 
years to be incorporated in the High-School 
course, geometry and algebra should be begun 
and frequently are begun. It would be a great 
gain if at this point the work in mathematics 
could be taken up by trained teachers of mathe- 
matics and continued without a break for six 
years under teachers of equal preparation 
working as members of the corps of instruction 
of the same institution. If, further, it should 
be found practicable in the development of 
the High-Schools to enlarge the course by the 
addition of a year or two, the mathematics usu- 
ally required in college would also be included 


1See p. 30 of Report, submitted July, 1899. 


Conclusion 121 


in the extended High-School course, thus col- 
lecting under one teaching force the entire 
course in mathematics (excepting arithmetic) 
usually required as part of a liberal education. 
It would seem promising for the work in 
mathematics if this were done whenever the 
High-Schools are equipped with well-prepared, 
enthusiastic teachers of mathematics compe- 
tent to take up this work. Perhaps the High- 
School of the future will have a course of eight 
years instead of four. 

Ouite a little can be done, moreover, before 
any organic change in the constitution of the 
school system, like that recom- whatcanbe 
mended by the Committee on Col- ‘onenow. 
lege Entrance Requirements, can be achieved. 
No exception will probably be taken to the 
statement that it would be a decided advance, 

a. If the teachers and administrators of the 
three classes of institutions now giving instruc- 
tion in mathematics were to come into much 
closer touch and make serious and systematic 
efforts to bring their work into better correla- 
tion; and 

b. If all those curricula now in force which 
are constructed on the same lines, and are al- 
ready in fact essentially the same, were also 
brought into working uniformity. 

It is gratifying to note that much progress is 
being made along these lines. It is facilitated 


122 Teaching of fMatbematics in Prussia 


by frequent joint conferences and formal or in- 
formal organizations of institutions for the pur- 
pose of harmonizing their work. If these op- 
portunities are cordially embraced, much can 
surely thus be done gradually to diminish the 
tremendous waste which our present system 
engenders. 

II. The Preparation and Status of the Teach- 
ers.—After what has already been said in the 
description of the Prussian system, it is almost 
superfluous to add that the excellence of the 
Prussian work in mathematics may be attrib- 
uted to the scientific and pedagogic prep- 
aration of the teachers for their work, more 
largely perhaps than to any other single cause. 
There must be taken into account, however, 
not simply the preparation of any one genera- 
tion, but the cumulative effect of generation 
after generation of thoroughly prepared teach- 
ers. The instruction which the pupil receives 
is a series of object lessons in teaching, and 
thus the well-taught pupil has a distinct advan- 
tage at the start, whereby the pedagogic meth- 
ods which one generation evolves are to some 
extent inherited without effort by the next. 

But this is only a beginning. The pupil’s 
Need of pedae POint of view and that of the teacher 
gogictrain- are very different. One views his ex- 
ion periences as a pupil from a pedagogic 
standpoint only through the vista of years. 


Conclusion 123 


After long trial the Germans have found that 
one year’s pedagogic preparation in the form 
of what may be called an apprentice year (the 
trial year) is insufficient, and have added an- 
other year preliminary to this—the seminary 
year, already described. The Prussian teacher 
has thus had two years of pedagogic preparation 
as well as three of scientific training in his spe- 
cialty when he enters the rank of the teachers, 
So long as the recruits to our teaching ranks 
include many with inadequate scientific prep- 
aration and with absolutely no pedagogic train- 
ing, we have no reason for disappointment if 
the comparison between our work and that 
done by a carefutly trained. army of teachers 
does not result in our favor. 

What can be done? Happily much zs being 
done. It is a most promising fact that the need 
for more thorough scientific and pedagogic 
preparation of our prospective teachers for 
their profession is widely felt, that the facilities 
offered for such preparation are constantly be- 
ing increased, and that the teachers, active and 
prospective, are availing themselves of these 
facilities in rapidly growing measure. Only a 
few decades since, all Americans who wished to 
study advanced mathematics were obliged to 
go to foreign lands; now ample opportunities 
for such study are offered from ocean to ocean 
in this land and are being eagerly embraced by 


124 Teaching of Matbematics in Prussia 


increasing numbers of students. With the due 
co-operation of the appointing authorities, the 
increasing supply of candidates who have had 
a more extended mathematical training will of 
itself raise the standard for the scientific equip- 
ment of the teachers of mathematics in our 
secondary schools, though it will doubtless re- 
quire a considerable number of years to reach 
the time when (as in Prussia) a large fraction 
of the teachers will be Doctors of Philosophy 
in mathematics, when all will have fulfilled the 
minimum time-requirement of advanced study 
for this degree, and when all will have had two 
years of training in the teaching of mathe- 
matics. Until this is the case, some disparity 
between the German work and our own must 
be expected. 

In the meanwhile, the Prussian results en- 
courage us to push on along the lines which 
we have already laid down. If there is one 
thing which the Prussian system teaches clear- 
ly, it is that scientific study (the subject-matter) 
and pedagogic training must both be given 
great weight in the preparation of the teacher. 
If either is slighted, the highest efficiency is 
not likely to be attained. 

The character of the problems with which 
Raisingof Mathematics deals, the methods which 
standards. it uses, and the faculties of the mind 
to which it appeals are such that the study and 


Conclusion 125 


the teaching of mathematics (and hence the 
preparation for teaching it) will be less affected 
by differences in national temperament and 
conditions than the teaching of any other sub- 
ject. What has proved successful in the prep- 
aration of the teachers of mathematics in one 
nation may be expected also to be successful 
under equally favorable circumstances in an- 
other nation. Still the introduction of any sys- 
tematic requirement of specific preparation for 
teachers is too closely connected with the prob- 
lems of general educational regulation and 
administration to be handled separately, and 
present progress with us is likely to be along 
the line of local raising of the standard for new 
appointment and voluntary strengthening of 
their equipment on the part of expectant teach- 
ers. Inthis connection, the third resolution of 
the Committee on College Entrance Require- 
ments’ should be mentioned: 


“III. RESOLVED: That the teachers in the 
secondary schools should be college graduates or 
have the equivalent of a college education.” 


In addition to the reasons given in support 
of this resolution in the report, the excellence 
of the work under the Prussian system may 
also be adduced. It is decidedly a step in the 


1 Report, p. 30. 


120 Teaching of Mathematics in Prussia 


right direction which it seems feasible to take 
now whenever new appointments are made. 

Some training of beginners in the teaching of 
mathematics may likewise be feasible now in 
institutions where there are one or more teach- 
ers with an aptitude for assisting and guiding 
the beginner in his first teaching. Whether 
this relation should be formal or informal, and 
whether the mathematical staff could hold with 
profit occasional or regular meetings for the 
discussion of pedagogic questions, are matters 
for local circumstances to determine. Under 
favorable conditions quite a little might be done 
in this way. 

However important the scientific and peda- 
gogic preparation of the teacher may be, it will 
Enthusiasm DPCar its ripest fruit only when ap- 
and devotion plied with devotion and enthusiasm 
panne throughout a lifetime. To this, the 
status of the Prussian teachers is especially con- 
ducive. To the German teacher his profession 
is a life-work, and not a stepping-stone or ad- 
junct to something else. Once fairly launched 
upon it, he almost never leaves it voluntarily, 
and his tenure of his position is assured for life, 
barring his own gross fault. He does not read law 
or practise medicine when free from the school 
duties which “keep the pot boiling ”’ until he is 
prepared to abandon the teacher’s vocation for 
one more lucrative or to his taste. On the con- 


Conclusion 127 


trary, he is fully alive to the fact that his teach- 
ing is a life-work, and that its honors and prizes 
will come mainly in the later portion. The ad- 
mirable salary and pension system frees him 
from harassing financial care and worry, and 
thus, with fully determined and assured official 
position and income (promotions coming in due 
course according to fixed regulations), he can 
give his entire thought and energy to the actual 
work of his profession. 

The result is that the young man is not in 
haste to realize on his few professional assets in 
his youth, before they shall have lost value to 
to him through his passing on to some more 
desirable position or occupation, but he steadily 
lays a deep and solid foundation for many years 
of future work. The traditions of the profes- 
sion, his relations with his colleagues, far and 
near, pedagogic meetings and conferences, the 
constant stream of publications which is flowing 
from these schools, all contribute to keeping 
enthusiasm and ambition alive. We Americans 
are not likely to overestimate the effect which 
the status of the Prussian teacher has upon the 
thoroughness and efficiency of his work. 

III. The Methods of Instruction.—The follow- 
ing points may be mentioned as deserving spe- 
cial attention under this head: 

1. Lhe Stress Latd on Classroom Work.—By far 
the larger part of the work as to quantity and 


128 Teaching of Mathematics in Prussia 


all of the work of acquisition, a//of the advance, 
all of the study of critical points, are done in 
Work under the class-exercise, under the direct 
instruction. ~oyidance and stimulus of the teacher. 
The German pupil receives from twenty-five to 
thirty hours per week of actual instruction, and 
his home-work (almost entirely of a drill or 
routine character) ranges from six to eighteen 
hours per week according to age. 

The advantage which the pupil gains from 
doing so large a fraction of his work in the class- 
exercise is of course dependent upon the 
teacher; the better the teacher, the more the 
pupil will gain from his instruction. The better 
the teacher, the better also he can utilize the 
time at his disposal. When the less well-pre- 
pared teacher must in the smaller fraction of 
time give such instruction as will enable the 
pupil to use the larger time of private study 
to best advantage, the results cannot but fall 
short of the best. 

One consequence of the quantity and charac- 
ter of the private work which the American 
pupil does is that he must receive a large 
portion of his instruction from a text-book. 
Whether this is desirable or not need not be 
discussed here; it is a fact that even with the 
best of teaching the pupil needs some guidance 
in his private work, and so long as so large a 
fraction of the American pupil’s work zs private 


Conclusion 129 


work a good text-book would seem to be the 
best substitute for the living teacher. 

In this connection the following recommen- 
dation of the Committee on College Entrance 
Requirements should be noted:! 


“XIV. RESOLVED: That we recommend an 
increase tn the school day in the secondary schools, 
to permit a larger amount of study in school under 
supervision.” 


Among the remarks in support of this reso- 
lution are the following: 

“In presenting this resolution the committee 
is aware that there is a great divergence in the 
length of the day in the secondary schools, the 
number and length of recitation periods, the 
noon intermissions, and the time devoted 
sacredly to study within the school-house. A 
few have two sessions, following the rule 
governing the elementary schools; some are 
from 8 A.M. to I P.M., and many from 9 A.M. 
to 2 P.M., with one-half hour a noon for a light 
lunch. 

We appreciate the almost unanimous and 
perhaps enlightened opposition on the part of 
teachers to the proposition for a longer school 
day. The committee believes, however, that 
it isa subject for intelligent discussion, and that 


1 Report, p. 40. 


130 Teaching of Mathematics in Prussia 


the weight of argument favors a longer day. 
The committee does not trace its convictions 
on this matter to the fact that the German sec- 
ondary schools are one-half longer in session 
than our schools and no hardship seems to 
result. 

There is no disposition to imitate European 
methods because they are European, but we 
believe it is easily demonstrable that it is in the 
class recitation and under the inspiration and 
instruction of the teacher, and not in study 
hours at home, that the pupil acquires the bulk 
of his scholastic knowledge.” 

Though the Committee justly bases its rec- 
ommendations upon broader ground than the 
fact that German schools have longer sessions 
than our schools, the scope of the present re- 
port is restricted precisely to a study of the 
German schools, and while the teaching of the 
German schools is along the same general line, 
it differs in some details from the recommenda- 
tions of the Committee. 

The important point is not so much that the 
German school day is longer (from 8 A.M. to 
A I P.M. is quite customary in Berlin), 
instruction as that the ¢zme of instruction is lon- 
needed. 5 : 

ger for each pupil. Such a thing 
as “study in school under supervision” is un- 
known in the Prussian schools. The pupil is ” 
participating in class-work, receiving instruc- 


Conclusion 131 


tion during the entire time he is at school. 
The hour-schedule is invariably arranged so 
that all the work of each class is consecutive, 
and the class is dismissed as soon as its instruc- 
tion is ended. Neither pupils nor teachers are 
expected to be present except when receiving 
or giving instruction. 

While doubtless the private study of the 
pupils will as a rule be more effective if done 
‘in the midst of all the paraphernalia incident 
to study,” actual instruction amid the same 
surroundings (with a corresponding diminution 
of the amount of study to be done in private) 
would be still more effective. To transfer study 
hours from the home to the school would be a 
gain, but to transform study hours into hours of 
instruction would be a far greater gain. This 
transformation would hardly arouse much op- 
position on the part of the teachers provided it 
did not entail additional burdens upon them. 

So far as Prussia is concerned, she does not 
secure the greater quantity of instruction by 
requiring her teachers to teach more hours, but 
by providing more teachers. The Prussian 
teacher teaches from twenty to twenty-two 
hours per week with a recess of ten minutes 
deducted from each hour. This will not com- 
pare very unfavorably with the amount of 
teaching done by American teachers in _the 
corresponding work. 


132 Ucaching of Mathematics in Prussia 


2. The Pause-—Of what may be called the 
minor features of German instruction, none per- 
haps is more productive of good results than the 
ten-minute pause at the close of each hour, 
bringing, as it does, complete relaxation for both 
pupils and teachers. It is not needful to dwell 
upon it further than this mention. The Ameri- 
can custom of passing from one subject to the 
next as quickly as the change can be effected, 
in the attempt to keep the minds of teachers 
and pupils on a tense strain for several hours at 
a time, is much to be deplored. It is not true 
economy to strive to have as few seconds as 
possible “lost” in the change from one subject 
of instruction to the next. 

It may not be amiss to add that of the 
thousands of German boys whom I saw during 
the pause not one was engaged in private study; 
I saw no hint of a feeling that these minutes 
might be used to put the last touches to the 
preparation for the work of the next hour. 

3. The Distribution of the Mathematical Work. 
—The three serious differences between the 
Three points German and the American distribu- 
ofdifference. tion of the mathematical work—zzz., 
the scope of arithmetic, the order in which the 
subjects are begun, and the rate at which they 
are continued—are no longer open questions in 
Germany. Experience has settled them _be- 
yond a peradventure. Arithmetic is restricted 


Conclusion 133 


to the most essential work of computation, 
demonstrative geometry is begun before alge- 
bra, and the study of each mathematical sub- 
ject extends over several years instead of being 
studied at high pressure during a much shorter 
time, as is done with us. The following table 
gives the number of years over which the study 
of each subject extends both in the German 
schools and in the American schools which 
have been brought into comparison with them 
above. In the American schools the subjects 
are kept quite distinct, and only one subject is 
taken up at a time. In the German schools 
the subject of study is mathematics, and its vari- 
ous subjects are developed side by side. At no 
time is only one subject being studied with the 
exception of the two lowest years (Sexta and 
Quinta), in which only arithmetic is taken up. 








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The consensus of American mathematico- 
pedagogic thought is strongly in favor of re- 
ducing the material taken up in arithmetic 
both by deferring topics to appropriate points 


wR 


134 Teaching of Mathematics in’ Prussia 


in algebra and geometry and by omitting others 
altogether. Reforms along this line have al- 
ready been made and others may confidently 
be expected. The time thus gained permits 
the earlier beginning of algebra and geometry. 
The door is also open for a change in the order 
of beginning these subjects, and for a change 
from the practice of forcing pupils to bolt them 
in a mass. Whether these changes are wise 
and feasible is another question. They cer- 
tainly seem wise from the stand-point of peda- 
gogic theory. No one, probably, denies that 
the beginnings of demonstrative geometry are 
more simple and concrete than those of alge- 
bra; unless, indeed, the latter is taught simply 
as mechanical routine, blind manipulation of 
symbols according to arbitrary rules, a thing 
above all others to be avoided, and which even 
without any encouragement is peculiarly tempt- 
ing to many beginners in algebra. 

It is likewise clear that the elements of alge- 
bra are simpler than the more difficult parts of 
Gradual geometry, that each of these subjects 
growth. can be of assistance to the other, and 
that a ripe understanding of them both is best 
attained if the work extends over._a number of 
years with gradual progress to the more diffi- 
cult portions as the pupil’s mind develops. If 
the young pupil with unformed and growing 
powers is to make progress with understanding 


Conclusion 135 


in mathematics, his acquisitions must have time 
to take root. The facts of algebra and geome- 
try may perhaps be acquired equally well in a 
more concentrated course; the pupil may be 
equally well prepared for an examination by 
taking up the subject relatively late and cov- 
ering it at a more rapid rate, but his grasp 
will hardly be as firm or as lasting as though 
he and the subject had grown up together. 

So much from the stand-point of pedagogic 
theory. What can be said from the practical 
stand-point? It is here that the German schools 
with their thorough trial of the plans just dis- 
cussed offer a reply of great value. They have 
tried the experiment for us. The question is 
practically the same there as here; differences 
of national temperament and conditions are not 
of serious consequence; the determination of 
these questions depends upon far more funda- 
mental and characteristic traits of the mind, 
common to all nations. While in no sense ac- 
cepting the German stamp as guarantee of ex- 
cellence, it would be equally foolish to go to 
the other extreme and refuse to consider solu- 
tions which the Germans have worked out for 
the same problems with which we are now 
grappling. The German higher-school system 
speaks with no uncertain tone of the practical 
advantages of beginning both geometry and 
algebra early, the former first, and continuing 


136 © Teaching of Mathematics in Prussia 


their study simultaneously throughout quite a 
number of years. With the pedagogic theory 
thus corroborated by practical experience, it 
would hardly be an experiment to interchange 
the order of algebra and geometry, and to 
carry on the subject simultaneously through a 
longer period. The practical difficulties in the 
way of simply interchanging the subjects are 
not very serious, but it would require consid- 
erable readjustment of programmes to put the 
second change into effect. Still the advantage 
to be gained is sufficiently considerable to war- 
rant strenuous efforts to overcome the difficul- 
ties which may be in the way. 

4. The Method of Instruction in Mathematics. 
—The manner of instruction in Prussia has 
The genetic already been described. It may be 
pete styled the genetic method. The sub- 
ject grows up in the class treatment, which 
does not require any previous preparation of 
the pupil for the work of the hour. A text- 
book may quite well be used with this method 
of instruction, but the pupil refers to it after 
the class treatment, not before. 

The genetic method has been little used 
in America, but there is a strong trend tow- 
ard the use of a somewhat similar method 
styled the heuristic method. This method has © 
been used hitherto almost exclusively in the 
teaching of geometry. It resembles the ge- 


Conclusion 137 


netic method in the marked effort it makes to 
keep the pupil thinking for himself. It differs 
from the genetic method in that the class is 
the working unit in the genetic method while 
the pupil is the working unit in the heuristic 
method. 

This difference may almost be said to be 
characteristic of the German and the American 
instruction in mathematics. In Ger- class ys. 
many the class works as a whole _ [mdlvidual. 
under the guidance of the instructor; when a 
pupil speaks it is part of a concerted action, as 
momentary spokesman for the class, as it were. 
In the American class the individual pupil 
demonstrates, explains, asks and answers ques- 
tions; the others listen to him. Whatever the 
style of instruction, the individual pupil is 
prominent. It is difficult to give a clear de- 
scription of the distinction, but it is readily felt 
by one who visits classes systematically in both 
nations, and the feeling grows stronger with 
the number and frequency of the visits. In 
this desire to give free play to the ability of in- 
dividuals the difference between the national 
temperament of the two peoples is quite appar- 
ent, and while we cannot cherish this desire 
too highly nor encourage too warmly every 
attempt to avoid hampering the growth of the 
individual, it must not be forgotten that the aim 
of the class-work is the greatest good of the 


138 Teaching of Mathematics in Prussia 


greatest number, and that perhaps this may not 
be attained by giving the pupils in turn indi- 
vidual prominence. 

The heuristic method does not permit the 
use of a text-book of the ordinary style; either 
The heuristic 2 Specially prepared text is used or 
mgthod: none. In the class-hour the pupil 
presents what he has already worked out, and 
the instructor helps him over the difficulties 
which he has not been able to surmount by 
himself. The genetic method is firmly estab- 
lished in Germany, and the heuristic method 
has been used with success in this country, 
where it has enthusiastic advocates. In either 
method a text if used plays a very minor part, 
and success depends upon the aptitude of the 
teacher to use the method. Teachers should 
not venture upon either of these methods with- 
out special training for their use. The best 
training that can be suggested for teachers 
already at work, who feel the impulse toward 
the heuristic method, often called “ teaching 
without a text,” is to seek to digest the subject 
in contradistinction from its presentation in any 
particular text, so that, whatever text may be 
used, it is the subject that is being taught. The 
time for the adoption of the heuristic method 
is when the teacher feels cramped by any other. 
The primary desideratum is that the teacher 
have athorough mastery of the subject, and this 


Conclusion 139 


requires that his mathematical horizon be much 
wider than what is actually taught. No teacher 
is ready to teach without a text who is not dis- 
tinctly conscious of teaching the subject even 
though teaching with a text. When this stage 
is reached, the use of a text is a matter of com- 
paratively little moment which the teacher’s 
preferences and local conditions will determine. 
In general, as long as the pupils work apart 
from the instructor so much of their time, as is 
the case with us, a good text may often be the 
best substitute available for the personal guid- 
ance of the teacher. 

It is not necessary to discuss the classroom 
methods of the Germans further here. They 
have been described in sufficient detail to en- 
able each reader to appreciate whatever they 
may contain that will be helpful or suggestive 
to him; what would be novel to one will be in 
daily use in the classroom of another. Our 
American teachers are eclectic, and have the 
freedom of a wide range of method; the good 
features of the Prussian classroom work may 
be seen, some here, some there, in many an 
American classroom to-day. It may therefore 
well be left to each reader to draw what spe- 
cific conclusions he may think best. If, how- 
ever, the tale should end with a moral, that 
moral would undoubtedly be this: 

The chief thing ts that every teacher be active 


140 Teaching of Mathematics in Prussia 


and growing scientifically and pedagogically ; that 
he add to his mathematical attainments year by 
year, and that he study to improve in the art of 
teaching. 

When this is the case throughout our land, all 
other needed developments will follow. 

Finally, the disparity which we have been 
considering seems to be fully accounted for by 
the less favorable circumstances under which 
the American work is done; in view of all the 
circumstances, we may well be gratified with 
what we have done and are doing, and I am con- 
fident that thoughtful consideration of the situ- 
ation here and elsewhere will justify the as- 
sured expectation that, with the attainment of 
equally favorable conditions in the various re- 
spects which have been discussed, the work 
done in American school mathematics need not 
fear the test of comparison. 

As this report is on the point of receiving its 
closing words, the article of Professor Miinster- 
berg, of Harvard, on the German school-work 
(Atlantic Monthly, May, 1900) comes to hand. 
Professor Miinsterberg writes from the point of 
view of one educated under the German sys- 
tem and learning to know the American system 
later in life. The present report is written from 
the point of view of one educated under the 
American system and learning to know the 
German system later in life. There is always 


Conclusion 141 


danger that a visitor will see the good points 
of a system, that its differences from the system 
under which he has been bred and whose short- 
comings he knows intimately, will impress him 
in the most favorable light possible, while the 
weak points of the system will probably be of 
a negative and less striking character and be 
appreciated fully only upon a much closer ac- 
quaintance with the system than a visitor ever 
gets. With this danger in mind, the writer has 
taken pains to exclude from this report mere 
Opinions and impressions which could not be 
substantiated by facts and figures, or supported 
by good reasons. It is an additional gratifica- 
tion to find that Professor Miinsterberg finds 
himself able, after completing the course of a 
gymnasium, to make even stronger statements 
as to the excellence of the work done in these 
institutions than any contained in this report. 
The words of advice which he gives to Amer- 
ican educators are especially pertinent at the 
present time and deserve most careful consid- 
eration. 





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